Theory of functions of a complex variable

•

ill

THEOBY OF FUNCTIONS

OF A

COMPLEX VARIABLE.

Itonlion: C. J. CLAY AND SONS,

CAMBEIDGE UNIVEESITY PEESS WAEEHOUSE,

AVE MAEIA LANE.

CAMBEIDGE : DEIGHTON, BELL, AKD CO.

LEIPZIG : F. A. BROCKHAUS.
NEW YORK: MACMILLAN AND CO.

THEOEY OF FUNCTIONS

OF A

COMPLEX VARIABLE

BY

A. R FORSYTE, So.D., F.RS.,

FELLOW OF TRINITY COLLEGE, CAMBRIDGE.

CAMBEIDGE:
AT THE UNIVERSITY PRESS.

1893

All rights reserved. .

Mtth. U. 01.

PRINTED BY C. J. CLAY, M.A. AND SONS,
AT THE UNIVERSITY PRESS.

PEEFACE.

AMONG the many advances in the progress of mathematical
XlL science during the last forty years, not the least remarkable
are those in the theory of functions. The contributions that are
still being made to it testify to its vitality : all the evidence points
to the continuance of its growth. And, indeed, this need cause no
surprise. Few subjects can boast such varied processes, based
upon methods so distinct from one another as are those originated
by Cauchy, by Weierstrass, and by Biemann. Each of these
methods is sufficient in itself to provide a complete development ;
combined, they exhibit an unusual wealth of ideas and furnish
unsurpassed resources in attacking new problems.

It is difficult to keep pace with the rapid growth of the
literature which is due to the activity of mathematicians,
especially of continental mathematicians : and there is, in con
sequence, sufficient reason for considering that some marshalling
of the main results is at least desirable and is, perhaps, necessary.
Not that there is any dearth of treatises in French and in
German : but, for the most part, they either expound the pro
cesses based upon some single method or they deal with the
discussion of some particular branch of the theory.

814033

PREFACE

The present treatise is an attempt to give a consecutive
account of what may fairly be deemed the principal branches of
the whole subject. It may be that the next few years will see
additions as important as those of the last few years : this account
would then be insufficient for its purpose, notwithstanding the
breadth of range over which it may seem at present to extend.
My hope is that the book, so far as it goes, may assist mathe
maticians, by lessening the labour of acquiring a proper knowledge
of the subject, and by indicating the main lines, on which recent
progress has been achieved.

No apology is offered for the size of the book. Indeed, if
there were to be an apology, it would rather be on the ground
of the too brief treatment of some portions and the omissions
of others. The detail in the exposition of the elements of several
important branches has prevented a completeness of treatment
of those branches : but this fulness of initial explanations is
deliberate, my opinion being that students will thereby become
better qualified to read the great classical memoirs, by the study
of which effective progress can best be made. And limitations of
space have compelled me to exclude some branches which other
wise would have found a place. Thus the theory of functions of
a real variable is left undiscussed : happily, the treatises of Dini,
Stolz, Tannery and Chrystal are sufficient to supply the omission.
Again, the theory of functions of more than one complex variable
receives only a passing mention ; but in this case, as in most
cases, where the consideration is brief, references are given
which will enable the student to follow the development to
such extent as he may desire. Limitation in one other direction
has been imposed : the treatise aims at dealing with the general
theory of functions and it does not profess to deal with special
classes of functions. I have not hesitated to use examples of
special classes : but they are used merely as illustrations of the
general theory, and references are given to other treatises for
the detailed exposition of their properties.

PREFACE Vll

The general method which is adopted is not limited so that
it may conform to any single one of the three principal inde
pendent methods, due to Cauchy, to Weierstrass and to Biemann
respectively : where it has been convenient to do so, I have
combined ideas and processes derived from different methods.

The book may be considered as composed of five parts.

The first part, consisting of Chapters I — VII, contains the
theory of uniform functions : the discussion is based upon power-
series, initially connected with Cauchy's theorems in integration,
and the properties established are chiefly those which are con
tained in the memoirs of Weierstrass and Mittag-Leffler.

The second part, consisting of Chapters VIII — XIII, contains
the theory of multiform functions, and of uniform periodic
functions which are derived through the inversion of integrals
of algebraic functions. The method adopted in this part is
Cauchy's, as used by Briot and Bouquet in their three memoirs
and in their treatise on elliptic functions : it is the method that
has been followed by Hermite and others to obtain the properties
of various kinds of periodic functions. A chapter has been
devoted to the proof of Weierstrass's results relating to functions
that possess an addition-theorem.

The third part, consisting of Chapters XIV — XVIII, contains
the development of the theory of functions according to the
method initiated by Biemann in his memoirs. The proof which
is given of the existence-theorem is substantially due to Schwarz ;
in the rest of this part of the book, I have derived great assist
ance from Neumann's treatise on Abelian functions, from Fricke's
treatise on Klein's theory of modular functions, and from many
memoirs by Klein.

The fourth part, consisting of Chapters XIX and XX, treats
of conformal representation. The fundamental theorem, as to the
possibility of the conformal representation of surfaces upon one
another, is derived from the existence-theorem : it is a curious fact
that the actual solution, which has been proved to exist in general,
F. b

Vlll PREFACE

has been obtained only for cases in which there is distinct
limitation.

The fifth part, consisting of Chapters XXI and XXII, contains
an introduction to the theory of Fuchsian or automorphic functions,
based upon the researches of Poincare and Klein : the discussion is
restricted to the elements of this newly-developed theory.

The arrangement of the subject-matter, as indicated in this
abstract of the contents, has been adopted as being the most
convenient for the continuous exposition of the theory. But the
arrangement does not provide an order best adapted to one who is
reading the subject for the first time. I have therefore ventured
to prefix to the Table of Contents a selection of Chapters that
will probably form a more suitable introduction to the subject for
such a reader ; the remaining Chapters can then be taken in an
order determined by the branch of the subject which he wishes
to follow out.

In the course of the preparation of this book, I have consulted
many treatises and memoirs. References to them, both general
and particular, are freely made : without making precise reserva
tions as to independent contributions of my own, I wish in this
place to make a comprehensive acknowledgement of my obligations
to such works. A number of examples occur in the book : most of
them are extracted from memoirs, which do not lie close to the
direct line of development of the general theory but contain
results that provide interesting special illustrations. My inten
tion has been to give the author's name in every case where a
result has been extracted from a memoir : any omission to do so
is due to inadvertence.

Substantial as has been the aid provided by the treatises and
memoirs to which reference has just been made, the completion of
the book in the correction of the proof-sheets has been rendered
easier to me by the unstinted and untiring help rendered by
two friends. To Mr William Burnside, M.A., formerly Fellow of

PREFACE

Pembroke College, Cambridge, and now Professor of Mathematics
at the Royal Naval College, Greenwich, I am under a deep debt
of gratitude : he has used his great knowledge of the subject in
the most generous manner, making suggestions and criticisms that
have enabled me to correct errors and to improve the book in
many respects. Mr H. M. Taylor, M. A., Fellow of Trinity College,
Cambridge, has read the proofs with great care : the kind assist
ance that he has given me in this way has proved of substantial
service and usefulness in correcting the sheets. I desire to
recognise most gratefully my sense of the value of the work which
these gentlemen have done.

It is but just on my part to state that the willing and active
co-operation of the Staff of the University Press during the pro
gress of printing has done much to lighten my labour.

It is, perhaps, too ambitious to hope that, on ground which
is relatively new to English mathematics, there will be freedom
from error or obscurity and that the mode of presentation in this
treatise will command general approbation. In any case, my aim
has been to produce a book that will assist mathematicians in
acquiring a knowledge of the theory of functions : in proportion
as it may prove of real service to them, will be my reward.

A. R. FORSYTE.

TRINITY COLLEGE, CAMBRIDGE.
25 February, 1893.

CONTENTS.

The following course is recommended, in the order specified, to those who are
reading the subject for the first time : The theory of uniform functions, Chapters
I— V ; Conformal representation, Chapter XIX ; Multiform functions and uniform
periodic functions, Chapters VIII— XI ; Riemanris surfaces, and Riemann's theory
of algebraic functions and their integrals, Chapters XIV— XVI, XVIII.

CHAPTER I.

GENERAL INTRODUCTION.

§§

PAGE

1—3. The complex variable and the representation of its variation by points

in a plane ,

4. Neumann's representation by points on a sphere ... 4

5. Properties of functions assumed known ... Q
6, 7. The idea of complex functionality adopted, with the conditions neces
sary and sufficient to ensure functional dependence ... 6

8. Riemann's definition of functionality ... g

9. A functional relation between two complex variables establishes the

geometrical property of conformal representation of their planes . 10

10, 11. Relations between the real and the imaginary parts of a function of z 11
12, 13. Definitions and illustrations of the terms monogenic, uniform, multiform,

branch, branch-point, holomorphic, zero, pole, meromorphic . . . 14

CHAPTER II.

INTEGRATION OF UNIFORM FUNCTIONS.

14, 15. Definition of an integral with complex variables ; inferences . . . ' 18
16. Proof of the lemma I I (^ - £ \ dxdy=\(pdx -\-qdy), under assigned

I \ fll'1 (it I I J •* J. •/ f ' O

conditions 21

CONTENTS

§§ PAGE

17, 18. The integral \f(z)dz round any simple curve is zero, when f(z) is

Cz

holomorphic within the curve; and I /(*)<& is a holomorphic

J a

function when the path of integration lies within the curve . . 23
19. The path of integration of a holomorphic function can be deformed

without changing the value of the integral ..... 26

20—22. The integral =— . I '— '- dz, round a curve enclosing a, is /(a) when
27rt J z — a

f(z) is a holomorphic function within the curve; and the integral

J_ [ /(*) dz is — ,—^. Superior limit for the modulus of
27rt J(z-a)n + 1 n\ dan

the nth derivative of /(a) in terms of the modulus of /(a) . . 27

23. The path of integration of a meromorphic function cannot be deformed

across a pole without changing the value of the integral. . . 34

24. The integral of any function (i) round a very small circle, (ii) round a

very large circle, (iii) round a circle which encloses all its infinities

and all its branch-points ......... 35

25. Special examples ............ ••

CHAPTER III.

EXPANSIONS OF FUNCTIONS IN SERIES OF POWERS.

26, 27. Cauchy's expansion of a function in positive powers of z - a ; with re
marks and inferences 43

28—30. Laurent's expansion of a function in positive and negative powers of

z - a ; with corollary 47

31. Application of Cauchy's expansion to the derivatives of a function . 51

32, 33. Definition of an ordinary point of a function, of the domain of an
ordinary point, of an accidental singularity, and of an essential
singularity .......••••• 52

34, 35. Continuation of a function by means of elements over its region of

continuity 54

36. Schwarz's theorem on symmetric continuation across the axis of real

quantities 57

CHAPTER IV.

UNIFORM FUNCTIONS, PARTICULARLY THOSE WITHOUT ESSENTIAL
SINGULARITIES.

37. A function, constant over a continuous series of points, is constant

everywhere in its region of continuity 59

38, 39. The multiplicity of a zero, which is an ordinary point, is finite; and

a multiple zero of a function is a zero of its first derivative . . 61

CONTENTS Xlll

§§ PAGE

40. A function, that is not a constant, must have infinite values . . 63

41, 42. Form of a function near an accidental singularity 64

43, 44. Poles of a function are poles of its derivatives ..... 66
45, 46. A function, which has infinity for its only pole and has no essential

singularity, is an algebraical polynomial ...... 69

47. Integral algebraical and integral transcendental functions ... 70

48. A function, all the singularities of which are accidental, is an algebraical

meromorphic function .......... 71

CHAPTER V.

TRANSCENDENTAL INTEGRAL UNIFORM FUNCTIONS.

49, 50. Construction of a transcendental integral function with assigned zeros
a1? a2, a3, ..., when an integer s can be found such that 2|an|~8

is a converging series 74

51. Weierstrass's construction of a function with any assigned zeros . . 77
52, 53. The most general form of function with assigned zeros and having

its single essential singularity at 0=00 . . . . . . 80

54. Functions with the singly-infinite system of zeros given by ;Z = TO<B, for

integral values of m 82

55 — 57. Weierstrass's o--function with the doubly-infinite system of zeros given

by z=ma> + m'a>, for integral values of TO and of TO' . . . . 84
58. A function cannot exist with a triply-infinite arithmetical system of zeros 88

59, 60. Class (genre) of a function 89

61. Laguerre's criterion of the class of a function 91

CHAPTER VI.

FUNCTIONS WITH A LIMITED NUMBER OF ESSENTIAL SINGULARITIES.

62. Indefiniteness of value of a function at an essential singularity . . 94

63. A function is of the form O { — =- ) + P (z — 6) in the vicinity of an essen-

\»— o/

tial singularity at b, a point in the finite part of the plane . . 96
64, 65. Expression of a function with n essential singularities as a sum of n

functions, each with only one essential singularity .... 99
66, 67. Product-expression of a function with n essential singularities and no

zeros or accidental singularities 101

68 — 71. Product-expression of a function with n essential singularities and with

assigned zeros and assigned accidental singularities ; with a note

on the region of continuity of such a function . . . .104

xiv CONTENTS

CHAPTER VII.

FUNCTIONS WITH UNLIMITED ESSENTIAL SINGULARITIES,
AND EXPANSION IN SERIES OF FUNCTIONS.

§§ 1>AGE

72. Mittag-Leffler's theorem on functions with unlimited essential singu

larities, distributed over the whole plane 112

73. Construction of subsidiary functions, to be terms of an infinite sum . 113
74_76. Weierstrass's proof of Mittag-Leffler's theorem, with the generalisation

of the form of the theorem 114

77, 78. Mittag-Leffler's theorem on functions with unlimited essential singu
larities, distributed over a finite circle 117

79. Expression of a given function in Mittag-Leffler's form .... 123

80. General remarks on infinite series, whether of powers or of functions . 126

81. A series of powers, in a region of continuity, represents one and only

one function ; it cannot be continued beyond a natural limit . . 128

82. Also a series of functions : but its region of continuity may consist of

distinct parts 129

83. A series of functions does not necessarily possess a derivative at points

on the boundary of any one of the distinct portions of its region

of continuity ........••• 133

84. A series of functions may represent different functions in distinct parts

of its region of continuity ; Tannery's series 136

85. Construction of a function which represents different assigned functions

in distinct assigned parts of the plane . . . . . .138

86. Functions with a line of essential singularity 139

87. Functions with an area of essential singularity or lacunary spaces . 141

88. Arrangement of singularities of functions into classes and species . . 146

CHAPTER VIII.

MULTIFORM FUNCTIONS.

89. Branch-points and branches of functions 149

90. Branches obtained by continuation: path of variation of independent

variable between two points can be deformed without affecting a
branch of a function if it be not made to cross a branch-point . 150

91. If the path be deformed across a branch-point which affects the branch,

then the branch is changed I55

92. The interchange of branches for circuits- round a branch-point is cyclical 156

93. Analytical form of a function near a branch-point 157

94. Branch-points of a function defined by an algebraical equation in their

relation to the branches : definition of algebraic function . . 161
95, 96. Infinities of an algebraic function 163

CONTENTS xv

PAGE

97. Determination of the branch-points of an algebraic function, and of the

cyclical systems of the branches of the function ... 168

98. Special case, when the branch-points are simple : their number . . 174

99. A function, with n branches and a limited number of branch-points and

singularities, is a root of an algebraical equation of degree n. . 175

CHAPTER IX.

PERIODS OF DEFINITE INTEGRALS, AND PERIODIC FUNCTIONS
IN GENERAL.

100. Conditions under which the path of variation of the integral of a

multiform function can be deformed without changing the value

of the integral ....... J§Q

101. Integral of a multiform function round a small curve enclosing a

branch-point ....... 183

102. Indefinite integrals of uniform functions with accidental singularities ;

fdz f dz

j i ' 2 ..... • ..... 184

103. Hermite's method of obtaining the multiplicity in value of an integral;

sections in the plane, made to avoid the multiplicity . . .185

104. Examples of indefinite integrals of multiform functions ; \wdz round

any loop, the general value of J(l - 22) ~ * dz, of J{1 - z2) (1 - k^}} ~ * dz,

and of 5{(z-el)(z-e2)(g-es)}-*dz ....... 189

105. Graphical representation of simply-periodic and of doubly-periodic

functions ....... 198

106. The ratio of the periods of a uniform doubly-periodic function is not

real ............. 200

107, 108. Triply-periodic uniform functions of a single variable do not exist . 202

109. Construction of a fundamental parallelogram for a uniform doubly-

periodic function ....... 205

110. An integral, with more periods than two, can be made to assume any

value by a modification of the path of integration between the
limits ........ 208

CHAPTER X.

SIMPLY-PERIODIC AND DOUBLY-PERIODIC FUNCTIONS.

2rrzi

111. Simply-periodic functions, and the transformation Z=e w . ;' • . 211

112. Fourier's series and simply-periodic functions 213

113, 114. Properties of simply-periodic functions without essential singularities

in the finite part of the plane 214

115. Uniform doubly-periodic functions, without essential singularities in

the finite part of the plane 218

116. Properties of uniform doubly-periodic functions 219

CONTENTS

§§

117. The zeros and the singularities of the derivative of a doubly-periodic

function of the second order . . 231

118, 119. Kelations between homoperiodic functions . . ... . • 233

CHAPTER XL

DOUBLY-PERIODIC FUNCTIONS OF THE SECOND ORDER.

120 121. Formation of an uneven function with two distinct irreducible in
finities; its addition-theorem 243

122, 123. Properties of Weierstrass's o-function . . . . • • 247
124. Introduction of f (2) and of Q(z) 250

125, 126. Periodicity of the function #> (z), with a single irreducible infinity of

degree two; the differential equation satisfied by the function #> (2) 251

127. Pseudo-periodicity of f(«) • 255

128. Construction of a doubly-periodic function in terms of f (z) and its

derivatives • . - . • • 256

129. On the relation qw'- j/eo = ±%iri 25>7

130. Pseudo-periodicity of a (z) • 259

131. Construction of a doubly-periodic function as a product of o-functions ;

with examples 259

132. On derivatives of periodic functions with regard to the invariants

#2 and £3 * ' ' lfK

133 135. Formation of an even function of either class 266

CHAPTER XII.

PSEUDO-PERIODIC FUNCTIONS.

136. Three kinds of pseudo-periodic functions, with the characteristic equa
tions 273

137, 138. Hermite's and Mittag-Leffler's expressions for doubly-periodic functions

of the second kind 275

139. The zeros and the infinities of a secondary function . . - . . 280

140, 141. Solution of Lamp's differential equation 281

142. The zeros and the infinities of a tertiary function .... 286

143. Product-expression for a tertiary function 287

144—146. Two classes of tertiary functions ; Appell's expressions for a function

of each class as a sum of elements 288

147. Expansion in trigonometrical series 293

148. Examples of other classes of pseudo-periodic functions . . . 295

CONTENTS Xvii

CHAPTER XIII.

FUNCTIONS POSSESSING AN ALGEBRAICAL ADDITION-THEOREM.
§§ PAGE

149. Definition of an algebraical addition-theorem 297

150. A function defined by an algebraical equation, the coefficients of

which are algebraical functions, or simply-periodic functions, or
doubly-periodic functions, has an algebraical addition-theorem . 297

151 — 154. A function possessing an algebraical addition-theorem is either
algebraical, simply-periodic or doubly-periodic, having in each
instance only a finite number of values for an argument . . 300

155, 156. A function with an algebraical addition-theorem can be defined by a
differential equation of the first order, into which the independent
variable does not explicitly enter 309

CHAPTER XIV.

CONNECTIVITY OF SURFACES.

157—159. Definitions of connection, simple connection, multiple connection, cross
cut, loop-cut . . . .'.•-., 312

160. Relations between cross-cuts and connectivity 315

161. Relations between loop-cuts and connectivity 320

162. Effect of a slit .321

163, 164. Relations between boundaries and connectivity 322

165. Lhuilier's theorem on the division of a connected surface into

curvilinear polygons 325

166. Definitions of circuit, reducible, irreducible, simple, multiple, compound,

reconcileable 327

167, 168. Properties of a complete system of irreducible simple circuits on a

surface, in its relation to the connectivity 328

169. Deformation of surfaces 332

170. Conditions of equivalence for representation of the variable . . 333

CHAPTER XV.

RIEMANN'S SURFACES.

171. Character and general description of a Riemann's surface . - .. •. 336

172. Riemann's surface associated with an algebraical equation . . .338

173. Sheets of the surface are connected along lines, called branch-lines . 338

174. Properties of branch-lines 340

175, 176. Formation of system of branch-lines for a surface ; with examples . 341

177. Spherical form of Riemann's surface . . . 34(5

XV111 CONTENTS

§§ PAGE

178. The connectivity of a Eiemann's surface 347

179. Irreducible circuits : examples of resolution of Riemann's surfaces

into surfaces that are simply connected 350

180, 181. General resolution of a Riemann's surface 353

182. A Riemann's %-sheeted surface when all the branch-points are simple 355

183, 184. On loops, and their deformation 356

185. Simple cycles of Clebsch and Gordan 359

186 — 189. Canonical form of Riemann's surface when all the branch -points are

simple, deduced from theorems of Luroth and Clebsch. . . 361

190. Deformation of the surface 365

191. Remark on uniform algebraical transformations 367

CHAPTER XVI.

ALGEBRAIC FUNCTIONS AND THEIR INTEGRALS.

192. Two subjects of investigation 368

193, 194. Determination of the most general uniform function of position on a

Riemann's surface .......... 369

195. Preliminary lemmas in integration on a Riemann's surface . . . 372

196, 197. Moduli of periodicity for cross-cuts in the resolved surface . . . 373

198. The number of linearly independent moduli of periodicity is equal to

the number of cross-cuts, which are necessary for the resolution

of the surface into one that is simply connected .... 378

199. Periodic functions on a Riemann's surface, with examples . . . 379

200. Integral of the most general uniform function of position on a

Riemann's surface . . . ; 387

201. Integrals, everywhere finite on the surface, connected with the equa

tion w*=S(z) 388

202 — 204. Infinities of integrals on the surface connected with the algebraical
equation f (w, z) = 0, when the equation is geometrically interpret-
able as the equation of a (generalised) curve of the nth order . 388

205, 206. Integrals of the first kind connected with/(w, z) = 0, Demg functions
that are everywhere finite : the number of such integrals, linearly
independent of one another : they are multiform functions . . 394

207, 208. Integrals of the second kind connected with f (w, z) = 0, being func
tions that have only algebraical infinities; elementary integral of
the second kind .......... 396

209. Integrals of the third kind connected with/(w, z) = 0, being functions

that have logarithmic infinities 400

210, 211. An integral of the third kind cannot have less than two logarithmic

infinities ; elementary integral of the third kind .... 401

CONTENTS

CHAPTER XVII.

SCHWARZ'S PROOF OF THE EXISTENCE-THEOREM.

§§ PAGE
212, 213. Existence of functions on a Riemann's surface; initial limitation of

the problem to the real parts u of the functions . ... . . 405

214. Conditions to which u, the potential function, is subject . . . 407

215. Methods of proof : summary of Schwarz's investigation . . . 408
216 — 220. The potential-function u is uniquely determined for a circle by the gene
ral conditions and by the assignment of finite boundary values . 410

221. Also for any plane area, on which the area of a circle can be con-

formally represented 423

222. Also for any plane area which can be obtained by a topological com

bination of areas, having a common part and each conformally
representable on the area of a circle 425

223. Also for any area on a Riemann's surface in which a branch-point

occurs ; and for any simply connected surface .... 428
224 — 227. Real functions exist on a Riemann's surface, everywhere finite, and

having arbitrarily assigned real moduli of periodicity . . . 430

228. And the number of the linearly independent real functions thus ob

tained is 2p ........... 434

229. Real functions exist with assigned infinities on the surface and

assigned real moduli of periodicity. Classes of functions of the
complex variable proved to exist on the Riemann's surface . . 435

CHAPTER XVIII.

APPLICATIONS OF THE EXISTENCE-THEOREM.

230. Three special kinds of functions on a Riemann's surface . . . 437
231 — 233. Relations between moduli of functions of the first kind and those of

functions of the second kind 439

234. The number of linearly independent functions of the first kind on a

Riemann's surface of connectivity 2/; + l is p

235. Normal functions of the first kind ; properties of their moduli .

236. Normal elementary functions of the second kind : their moduli .
237, 238. Normal elementary functions of the third kind : their moduli : inter
change of arguments and parametric points 449

239. The inversion-problem for functions of the first' kind .... 453

240. Algebraical functions on a Riemann's surface without infinities at the

branch-points but only at isolated ordinary points on the surface :
Riemann-Roch's theorem : the smallest number of singularities
that such functions may possess . . . . . . .457

241. A class of algebraic functions infinite only at branch-points . . 460

242. Fundamental equation associated with an assigned Riemann's surface 462

XX CONTENTS

§§ PAGE

243. Appell's factorial functions on a Riemann's surface : their multipliers

at the cross-cuts 464

244, 245. Expression for a factorial function with assigned zeros and assigned
infinities; relations between zeros and infinities of a factorial
function . . ... . . .' . • • • 466

246. Functions defined by differential equations of the form / ( w, -y- ) = 0 470

\ **&)

247 — 249. Conditions that the function should be a uniform function of z. . 471
250, 251. Classes of uniform functions that can be so defined, with criteria of

discrimination . . • , • • • • • • • • 476

(dw\s
~T~ ) =/ (w) .... 482
az j

CHAPTER XIX.

CONFORMAL REPRESENTATION : INTRODUCTORY.

253. A relation between complex variables is the most general relation that

secures conformal similarity between two surfaces .... 491

254. One of the surfaces for conformal representation may, without loss of

generality, be taken to be a plane 495

255, 256. Application to surfaces of revolution ; in particular, to a sphere, so

as to obtain maps .......... 496

257. Some examples of conformal representation of plane areas, in par

ticular, of areas that can be conformally represented on the area

of a circle 501

258. Linear homographic transformations (or substitutions) of the form

w = ,: their fundamental properties 512

cz + d

259. Parabolic, elliptic, hyperbolic and loxodromic substitutions . . . 517

260. An elliptic substitution is either periodic or infinitesimal : substitu

tions of the other classes are neither periodic nor infinitesimal . 521

261. A linear substitution can be regarded geometrically as the result of

an even number of successive inversions of a point with regard

to circles . ......... 523

CHAPTER XX.

CONFORMAL REPRESENTATION : GENERAL THEORY.

262. Riemann's theorem on the conformal representation of a given area

upon the area of a circle with unique correspondence . . . 525

263, 264. Proof of Riemann's theorem : how far the functional equation is

algebraically determinate 526

265, 266. The method of Beltrami and Cayley for the construction of the

functional equation for an analytical curve 530

CONTENTS XXI

§§ PAGE

267, 268. Conformal representation of a convex rectilinear polygon upon the

half-plane of the variable 537

269. The triangle, and the quadrilateral, conformally represented . . 543

270. A convex curve, as a limiting case of a polygon .... 548
271, 272. Conformal representation of a convex figure, bounded by circular arcs :

the functional relation is connected with a linear differential
equation of the second order ........ 549

273. Conformal representation of a crescent ....... 554

274 — 276. Conformal representation of a triangle, bounded by circular arcs . 555
277 — 279. Relation between the triangle, bounded by circles, and the stereographic

projection of regular solids inscribed in a sphere .... 563

280. On families of plane algebraical curves, determined as potential-curves
by a single parameter u + vi : the forms of functional relation

), which give rise to such curves .... 575

CHAPTER XXI.

GROUPS OF LINEAR SUBSTITUTIONS.

281. The algebra of group-symbols 582

282. Groups, which are considered, are discontinuous and have a finite

number of fundamental substitutions 584

283, 284. Anharmonic group : group for the modular-functions, and division of

the plane of the variable to represent the group .... 586
285, 286. Fuchsian groups : division of plane into convex curvilinear polygons :

polygon of reference 591

287. Cycles of angular points in a curvilinear polygon .... 595

288, 289. Character of the division of the plane : example .... 599

290. Fuchsian groups which conserve a fundamental circle . . . 602

291. Essential singularities of a group, and of the automorphic functions

determined by the group ........ 605

292, 293. Families of groups : and their class 606

294. Kleinian groups : the generalised equations connecting two points in

space "... . . .609

295. Division of plane and division of space, in connection with Kleinian

groups 613

296. Example of improperly discontinuous group 615

CHAPTER XXII.

AUTOMORPHIC FUNCTIONS.

297. Definition of automorphic functions . . . . . '- . .619

298. Examples of functions, automorphic for finite discrete groups of sub

stitutions 620

299. Cayley's analytical relation between stereographic projections of posi

tions of a point on a rotated sphere 620

XX11 CONTENTS

§§ PAGE

300. Polyhedral groups ; in particular, the dihedral group, and the tetra-

hedral group 623

301, 302. The tetrahedral functions, and the dihedral functions . . . 628

303. Special illustrations of infinite discrete groups, from the elliptic

modular-functions 633

304. Division of the plane, and properties of the fundamental polygon of

reference, for any infinite discrete group that conserves a funda
mental circle ........... 637

305, 306. Construction of Thetafuchsian functions, pseudo-automorphic for an

infinite group of substitutions 641

307. Relations between the number of irreducible zeros and the number

of irreducible poles of a pseudo-automorphic function, constructed

with a rational algebraical meromorphic function as element . 645

308. Construction of automorphic functions ....... 650

309. The number of irreducible points, for which an automorphic function

acquires an assigned value, is independent of the value . . 651

310. Algebraical relations between functions, automorphic for a group :

application of Riemanu's theory of functions .... 653

311. Connection between automorphic functions and linear differential

equations ; with illustrations from elliptic modular-functions . 654

GLOSSARY OF TECHNICAL TERMS . . . . ; . . ' , . 659
INDEX OF AUTHORS QUOTED . . . . - . . . . . 662
GENERAL INDEX 664

CHAPTER I.

GENERAL INTRODUCTION.

1. ALGEBRAICAL operations are either direct or inverse. Without
entering into a general discussion of the nature of irrational and of imaginary
quantities, it will be sufficient to point out that direct algebraical operations
on numbers that are positive and integral lead to numbers of the same
character; and that inverse algebraical operations on numbers that are
positive and integral lead to numbers, which may be negative or fractional
or irrational, or to numbers which may not even fall within the class of real
quantities. The simplest case of occurrence of a quantity, which is not
real, is that which arises when the square root of a negative quantity is
required.

Combinations of the various kinds of quantities that may occur are of
the form x + iy, where x and y are real and i, the non-real element of the
quantity, denotes the square root of - 1. It is found that, when quantities
of this character are subjected to algebraical operations, they always lead to
quantities of the same formal character; and it is therefore inferred that
the most general form of algebraical quantity is x + iy.

Such a quantity ic + iy, for brevity denoted by z, is usually called a
complex variable*; it therefore appears that the complex variable is the
most general form of algebraical quantity which obeys the fundamental laws
of ordinary algebra.

2. The most general complex variable is that, in which the constituents
x and y are independent of one another and (being real quantities) are
separately capable of assuming all values from - oo to + oo ; thus a doubly-
infinite variation is possible for the variable. In the case of a real variable,
it is convenient to use the customary geometrical representation by measure
ment of distance along a straight line; so also in the case of a complex

* The conjugate complex, viz. x - iy, is frequently denoted by za.
F.

2 GEOMETRICAL REPRESENTATION OF [2.

variable, it is convenient to associate a geometrical representation with
the algebraical expression ; and this is the well-known representation of
the variable ac + iy by means of a point with coordinates x and y referred
to rectangular axes*. The complete variation of the complex variable z
is represented by the aggregate of all possible positions of the associated
point, which is often called the point z ; the special case of real variables
being evidently included in it because, when y = 0, the aggregate of
possible points is the line which is the range of geometrical variation of
the real "variable.

• The variation of z is said to be continuous when the variations of x and y
are contiguous. Continuous variation of z between two given values will
thus be represented by continuous variation in the position of the point z,
that is, by a continuous curve (not necessarily of continuous curvature)
between the points corresponding to the two values. But since an infinite
number of curves can be drawn between two points in a plane, continuity of
line is not sufficient to specify the variation of the complex variable ; and, in
order to indicate any special mode of variation, it is necessary to assign,
either explicitly or implicitly, some determinate law connecting the variations
of x and y or, what is the same thing, some determinate law connecting x
and y. The analytical expression of this law is the equation of the curve
which represents the aggregate of values assumed by the variable between
the two given values.

In such a case the variable is often said to describe the part of the curve
between the two points. In particular, if the variable resume its initial
value, the representative point must return to its initial position ; and then
the variable is said to describe the whole curve -f-.

When a given closed curve is continuously described by the variable,
there are two directions in which the description can take place. From
the analogy of the description of a straight line by a point representing a
real variable, one of these directions is considered as positive and the other

* This method of geometrical representation of imaginary quantities, ordinarily assigned to
Gauss, was originally developed by Argand who, in 1806, published his " Essai sur une maniere
de representer les quantites imaginaires dans les constructions geometriques." This tract was
republished in 1874 as a second edition (Gauthier-Villars) ; an interesting preface is added
to it by Hoiiel, who gives an account of the earlier history of the publications associated with
the theory.

Other references to the historical development are given in Chrystal's Text-book of Algebra,
vol. i, pp. 248, 249; in Holzmiiller's Einfilhrung in die Theorie der isogonalen Venvandschaften
und dcr conformen Abbildungen, verbunden mit Anwendungen auf mathematische Physik, pp. 1 — 10,
21 — 23 ; in Schlomilch's Compendium der hoheren Analysis, vol. ii, p. 38 (note) ; and in Casorati,
Teorica delle funzioni di variabili complesse, only one volume of which was published. In this
connection, an article by Cayley (Quart. Journ. of Math,, vol. xxii, pp. 270 — 308) may be
consulted with advantage.

t In these elementary explanations, it is unnecessary to enter into any discussion of
the effects caused by the occurrence of singularities in the curve.

2-]

THE COMPLEX VARIABLE

Fig. 1.

as negative. The usual convention under which one of the directions is
selected as the positive direction depends upon the conception that the curve
is the boundary, partial or complete, of some area ; under it, that direction is
taken to be positive which is such that the bounded area lies to the left of
the direction of description. It is easy to see that the same direction is taken
to be positive under an equivalent convention
which makes it related to the normal drawn
outwards from the bounded area in the same
way as the positive direction of the axis of y
is to the positive direction of the axis of x
in plane coordinate geometry.

Thus in the figure (fig. 1), the positive
direction of description of the outer curve
for the area included by it is DEF; the
positive direction of description of the inner
curve for the area without it (say, the area
excluded by it) is AGB ; and for the area
between the curves the positive direction of description of the boundary,
which consists of two parts, is DEF, ACB.

3. Since the position of a point in a plane can be determined by means
of polar coordinates, it is convenient in the discussion of complex variables
to introduce two quantities corresponding to polar coordinates.

In the case of the variable z, one of these quantities is (#2 + yn-)l, the
positive sign being always associated with it ; it is called the modulus* of
the variable and it is denoted, sometimes by mod. z, sometimes by \z .

The other is 0, the angular coordinate of the point z ; it is called the
argument (and, less frequently, the amplitude) of the variable. It is
measured in the trigonometrically positive sense, and is determined by
the equations

<K=\Z\ cos 6, y= z\ sin#,

so that z= z\eei. The actual value depends upon the way in which the
variable has acquired its value ; when variation
of the argument is considered, its initial value
is usually taken to lie between 0 and 2?r or, less
frequently, between -TT and +TT.

As z varies in position, the values of \z\
and 6 vary. When z has completed a positive
description of a closed curve, the modulus of z
returns to the initial value whether the origin Fig. 2.

Der absolute Metro,;) is often used by German writers.

1—2

GREAT VALUES OF

[3.

be without, within or on the curve. The argument of z resumes its initial
value, if the origin 0' (fig. 2) be without the curve ; but, if the origin 0 be
within the curve, the value of the argument is increased by 2-rr when z
returns to its initial position.

If the origin be on the curve, the argument of z undergoes an abrupt
change by TT as z passes through the origin ; and the change is an increase
or a decrease according as the variable approaches its limiting position on the
curve from without or from within. No choice need be made between these
alternatives; for care is always exercised to choose curves which do not
introduce this element of doubt.

4. Representation on a plane is obviously more effective for points at a
finite distance from the origin than for points at a very great distance.

One method of meeting the difficulty of representing great values is to
introduce a new variable z1 given by z'z=\\ the part of the new plane for
z which lies quite near the origin corresponds to the part of the old plane
for z which is very distant. The two planes combined give a complete
representation of variation of the complex variable.

Another method, in many ways more advantageous, is as follows. Draw
a sphere of unit diameter, touching the 2-plane at the origin 0 (fig. 3) on
the under side: join a point z in the plane to 0', the other extremity of
the diameter through 0, by a straight line cutting the sphere in Z.
Then Z is a unique representative of z, that is, a single point on the
sphere corresponds to a single point on the plane : and therefore the variable
can be represented on the surface of the sphere. With this mode of

Fig. 3.

representation, 0' evidently corresponds to an infinite value of z : and points
at a very great distance in the 2-plane are represented by points in the
immediate vicinity of 0' on the sphere. The sphei-e thus has the advantage
of putting in evidence a part of the surface jn which the variations of

4.] THE COMPLEX VARIABLE 5

great values of z can be traced*, and of exhibiting the uniqueness of
z — oo as a value of the variable, a fact that is obscured in the represen
tation on a plane.

The former method of representation can be deduced by means of the
sphere. At 0' draw a plane touching the sphere : and let the straight line
OZ cut this plane in z'. Then z is a point uniquely determined by Z
and therefore uniquely determined by z. In this new /-plane take axes
parallel to the axes in the 2-plane.

The points z and / move in the same direction in space round 00'
as an axis. If we make the upper side of the 2-plane correspond to the
lower side of the /-plane, and take the usual positive directions in the
planes, being the positive trigonometrical directions for a spectator looking
at the surface of the plane in which the description takes place, we have
these directions indicated by the arrows at 0 and at 0' respectively, so
that the senses of positive rotations in the two planes are opposite in
space. Now it is evident from the geometry that Oz and O'z' are
parallel ; hence, if 0 be the argument of the point z and & that of the
point z so that 6 is the angle from Ox to Oz and 6' the angle from O'x'
to O'z, we have

6 + ff = ZTT.

Oz 00'
Further, by similar triangles, -^-t = ^-f ,

that is, Oz . O'z' = OO'2 = 1.

Now, if z and z' be the variables, we have

z=0z.eei, z'=0'z'.effi,
so that zz'=0z.0'z' .e^s'^

= 1,
which is the former relation.

The /-plane can therefore be taken as the lower side of a plane touching
the sphere at 0' when the 2-plane is the upper side of a plane touching
it at 0. The part of the 2-plane at a very great distance is represented on
the sphere by the part in the immediate vicinity of 0' : and this part of
the sphere is represented on the /-plane by its portion in the immediate
vicinity of 0', which therefore is a space wherein the variations of infinitely
great values of z can be traced.

But it need hardly be pointed out that any special method of represent
ation of the variable is not essential to the development of the theory of
functions ; and, in particular, the foregoing representation of the variable,
when it has very great values, merely provides a convenient method of
dealing with quantities that tend to become infinite in magnitude.

* This sphere is sometimes called Neumann's sphere; it is used by him for the representation
of the complex variable throughout his treatise Vorlesungen uber Riemann'a Theorie der AlcVschen
Integrate (Leipzig, Teubner, '2nd edition, 1884).

6 CONDITIONS OF [5.

5. The simplest propositions relating to complex variables will be
assumed known. Among these are, the geometrical interpretation of opera
tions such as addition, multiplication, root-extraction ; some of the relations
of complex variables occurring as roots of algebraical equations with real
coefficients; the elementary properties of functions of complex variables
which are algebraical and integral, or exponential, or circular functions;
and simple tests of convergence of infinite series and of infinite products*.

6. All ordinary operations effected on a complex variable lead, as already
remarked, to other complex variables; and any definite quantity, thus
obtained by operations on z, is necessarily a function of z.

But if a complex variable w be given as a complex function of x
and y without any indication of its source, the question as to whether
w is or is not a function of z requires a consideration of the general idea
of functionality.

It is convenient to postulate u + iv as a form of the complex variable w,
where u and v are real. Since w is initially unrestricted in variation, we
may so far regard the quantities u and v as independent and therefore as
any functions of x and y, the elements involved in z. But more explicit
expressions for these functions are neither assigned nor supposed.

The earliest occurrence of the idea of functionality is in connection with
functions of real variables ; and then it is coextensive with the idea of
dependence. Thus, if the value of X depends on that of x and on no other
variable magnitude, it is customary to regard X as a function of x\ and
there is usually an implication that X is derived from x by some series of
operations^.

A detailed knowledge of z determines x and y uniquely ; hence the values
of u and v may be considered as known and therefore also w. Thus the
value of w is dependent on that of z, and is independent of the values
of variables unconnected with z; therefore, with the foregoing view of
functionality, w is a function of z.

It is, however, equally consistent with that view to regard w as a complex
function of the two independent elements from which z is constituted ; and
we are then led merely to the consideration of functions of two real
independent variables with (possibly) imaginary coefficients.

* These and other introductory parts of the subject are discussed in Chrystal's Text-book of
Algebra and in Hobson's Treatise on Plane Trigonometry.

They are also discussed at some length in the recently published translation, by G. L.
Cathcart, of Harnack's Elements of the differential and integral calculus (Williams and Norgate,
1891), the second and the fourth books of which contain developments that should be consulted
in special relation with the first few chapters of the present treatise.

These books, together with Neumann's treatise.cited in the note on p. 5, will hereafter be cited
by the names of their respective authors.

t It is not important for the present purpose to keep in view such mathematical expressions
as have intelligible meanings only when the independent variable is confined within limits.

6.] FUNCTIONAL DEPENDENCE 7

Both of these aspects of the dependence of w on z require that z be
regarded as a composite quantity involving two independent elements which
can be considered separately. Our purpose, however, is to regard z as the
most general form of algebraical variable and therefore as an irresoluble
entity ; so that, as this preliminary requirement in regard to z is unsatisfied,
neither of the aspects can be adopted.

7. Suppose that w is regarded as a function of z in the sense that it can
be constructed by definite operations on z regarded as an irresoluble
magnitude, the quantities u and v arising subsequently to these operations
by the separation of the real and the imaginary parts when z is replaced by
x + iy. It is thereby assumed that one series of operations is sufficient for
the simultaneous construction of u and v, instead of one series for u and
another series for v as in the general case of a complex function in § 6.
If this assumption be justified by the same forms resulting from the two
different methods of construction, it follows that the two series of opera
tions, which lead in the general case to u and to v, must be equivalent to
the single series and must therefore be connected by conditions ; that is, u
and v as functions of a; and y must have their functional forms related.

We thus take

u + iv — w = f(z) = f(x + iy)

without any specification of the form of f. When this postulated equation

is valid, we have

dw dw dz ,. , . dw

_ — . _ _ - I { 2/9 TTT _

dx dz dx J ^ ' dz'
dw _ dw "dz _ .,,. . . dw
frj = ~fad~y~V (Z) lfa'

• dw 1 dw dw

and therefore — = -— = —- ........................... (1)

dx i dy dz

equations from which the functional form has disappeared. Inserting the
value of w, we have

whence, after equating real and imaginary parts,

dv _du du _ dv
dx dy' dx dy"
These are necessary relations between the functional forms of u and v.

These relations are easily seen to be sufficient to ensure the required
functionality. For, on taking w = ii + iv, the equations (2) at once lead to

dw _ 1 dw
dx i dy '

,, , . dw .dw

that is, to -- — \- 1 — - = 0,

ox dy

8 RIEMANN'S [7.

a linear partial differential equation of the first order. To obtain the most
general solution, we form a subsidiary system

dx _ dy _ dw
T==T ==~0~*

It possesses the integrals w, x + iy; and then from the known theory of
such equations we infer that every quantity w satisfying the equation can be
expressed as a function of x + iy, i.e., of z. The conditions (2) are thus
proved to be sufficient, as well as necessary.

8. The preceding determination of the necessary and sufficient conditions
of functional dependence is based upon the existence of a functional form ;
and yet that form is not essential, for, as already remarked, it disappears from
the equations of condition. Now the postulation of such a form is equivalent
to an assumption that the function can be numerically calculated for each
particular value of the independent variable, though the immediate expres
sion of the assumption has disappeared in the present case. Experience of
functions of real variables shews that it is often more convenient to use
their properties than to possess their numerical values. This experience is
confirmed by what has preceded. The essential conditions of functional
dependence are the equations (1), and they express a property of the function

w, viz., that the value of the ratio -r is the same as that of ~- , or, in other

words, it is independent of the manner in which dz ultimately vanishes by
the approach of the point z + dz to coincidence with the point z. We are
thus led to an entirely different definition of functionality, viz. :

A complex quantity w is a function of another complex quantity z, when
they change together in such a manner that the value of -, is independent of
the value of the differential element dz.

This is Riemann's definition* ; we proceed to consider its significance.

We have

dw du + idv
dz dx + idy

/du .dv\ dx /du .dv\ du

__ I __ I n _ I _ _____ I I __ L ^ __ I _ Y. _

~~ \dx dxj dx + idy \dy dy/ dx + idy '
Let </> be the argument of dz ; then

_

cos <£ + 1 sin </>

* Ges. Werke, p. 5; a modified definition is adopted by him, ib., p. 81.

8.] DEFINITION OF A FUNCTION

and therefore

dw . (du .dv .du dv) „,. {du .dv .du dv

I I I i n I I I 1 a—^4> I J I t In

7 — i i«i T" ^ • 5 • 57 f • a** i<5 Tfc ^~ ~r * « «

«£ • [da; dx dy dy} (dx dx dy dy

Since -j— is to be independent of the value of the differential element dz,
dz

it must be independent of <f> the argument of dz ; hence the coefficient
of e-2*« in the preceding expression must vanish, which can happen only if

du _dv dv _ du
dx dy' dx dy "

These are necessary conditions; they are evidently also sufficient to make
^— independent of the value of dz and therefore, by the definition, to secure
that w is a function of z.

By means of the conditions (2), we have

dw _ du .dv _dw
dz dx dx dx '

dw .du dv 1 dw
and also — - = — i - — [_=_.

dz dy dy i dy

agreeing with the former equations (1) and immediately derivable from the
present definition by noticing that dx and idy are possible forms of dz.

It should be remarked that equations (2) are the conditions necessary
and sufficient to ensure that each of the expressions

udx — vdy and vdx + udy

is a perfect differential — a result of great importance in many investigations
in the region of mathematical physics.

When the conditions (2) are expressed, as is sometimes convenient, in
terms of derivatives with regard to the modulus of z, say r, and the
argument of z, say 0, they take the new forms

du_ldv dv _ Idu,

^ — ~ 57j > ^~ — ^TT. (^)-

or r dv or r da

We have so far assumed that the function has a differential coefficient —
an assumption justified in the case of functions which ordinarily occur. But
functions do occur which have different values in different regions of the

.z-plane, and there is then a difficulty in regard to the quantity ,W at the

boundaries of such regions ; and functions do occur which, though themselves
definite in value in a given region, do not possess a differential coefficient at
all points in that region. The consideration of such functions is not of
substantial importance at present : it belongs to another part of our subject.

10 CONFORMAL [8.

It must not be inferred that, because -j- is independent of the direction
in which dz vanishes when w is a function of z, therefore -=- has only one

value. The number of its values is dependent on the number of values of w :
no one of its values is dependent on dz.

A quantity, defined as a function by Riemann on the basis of this
property, is sometimes* called an analytical function; but it seems pre
ferable to reserve the term analytical in order that it may be associated
hereafter (§ 34) with an additional quality of the functions.

9. The geometrical interpretation of complex variability leads to impor
tant results when applied to two variables w and z which are functionally
related.

Let P and p be two points in different planes, or in different parts of
the same plane, representing w and z respectively; and suppose that P and
p are at a finite distance from the points (if any) which cause discontinuity
in the relationship. Let q and r be any two other points, z + dz and z + 8z,
in the immediate vicinity of p ; and let Q and E be the corresponding
points, w + dw and w + &w, in the immediate vicinity of P. Then

dw j ^ dw ?
dw = ^r- dz. bw = -r— of,
dz dz

the value of ~ being the same for both equations, because, as w is a function
dz

of z, that quantity is independent of the differential element of z. Hence

8w _ Bz
dw dz'

on the ground that , is neither zero nor infinite at z, which is assumed not

CL2

to be a point of discontinuity in the relationship. Expressing all the differ
ential elements in terms of their moduli and arguments, let

dz = a-eei, dw — rje^1,

Sz = oV'*, 8w = i)<$\
and let these values be substituted in the foregoing relation ; then

77' tr

tj a

$-$ = &-&.

Hence the triangles QPR and qpr are similar to one another, though
not necessarily similarly situated. Moreover the directions originally chosen
for pq and pr are quite arbitrary. Thus it appears that a functional relation

* Harnack, § 84.

1 11 V - (<M\* a. i^v

• I <a I — I "5 / ' \ ">

$a?/ \oyj \dy

9.] REPRESENTATION OF PLANES 11

between two complex variables establishes the similarity of the corresponding
infinitesimal elements of those parts of two planes which are in the immediate
vicinity of the points representing the two variables.

The magnification of the w-plane relative to the ^-plane at the corre
sponding points P and p is the ratio of two corresponding infinitesimal

lengths, say of QP and qp. This is the modulus of -^— ; if it be denoted by

m, we have

2 _ dw 2
dz

_ du dv du dv
dx dy dy dx '

Evidently the quantity m, in general, depends on the variables and
therefore it changes from one point to another ; hence the functional relation
between w and z does not, in general, establish similarity of finite parts of
the two planes corresponding to one another through the relation.

It is easy to prove that w = az + b, where a and b are constants, is the
only relation which establishes similarity of finite parts ; and that, with this
relation, a must be a real constant in order that the similar parts may be
similarly situated.

If u + iv = w = <}> (z), the curves u = constant and v = constant cut at
right angles; a special case of the proposition that, if <£ (x + iy) = u + v^,
where A, is a real constant and u, v are real, then u= constant and v= constant
cut at an angle X.

The process, which establishes the infinitesimal similarity of two planes
by means of a functional relation between the variables of the planes, may be
called the conformal representation of one plane on another*.

The discussion of detailed questions connected with the conformal representation is
deferred until the later part of the treatise, principally in order to group all such
investigations together ; but the first of the two chapters, devoted to it, need not be
deferred so late and an immediate reading of some portion of it will tend to simplify
many of the explanations relative to functional relations as they occur in the early
chapters of this treatise.

10. The analytical conditions of functionality, under either of the
adopted definitions, are the equations (2). From them it at once follows that

8^ + ty* = '

* By Gauss (Ges. Werke, t. iv, p. 262) it was styled conforme Abbildung, the name
universally adopted by German mathematicians. The French title is representation conforme ;
and, in England, Cayley has used orthomorphosis or ortliomorphic transformation.

12 CONDITIONS OF FUNCTIONAL DEPENDENCE [10.

so that neither the real nor the imaginary part of a complex function can be
arbitrarily assumed.

If either part be given, the other can be deduced ; for example, let u be
given ; then we have

7 j j

dv = ^-dx + — dy

dx dy

du , du j
= -=-dx+~-dy,
dy ox '

and therefore, except as to an additive constant, the value of v is

[i 9w 7 du -, \

- — dx + 5- dy I .
A dy ax ° I

In particular, when u is an integral function, it can be resolved into the
sum of homogeneous parts

MI + w2 + w3 + . . . ;

and then, again except as to an additive constant, v can similarly be
expressed in the form

Vl + V2 + V3 + ----

It is easy to prove that

dum dum

™>» = y-te-*-ty>

by means of which the value of v can be obtained.

The case, when u is homogeneous of zero dimensions, presents no
difficulty ; for we then have

v = c-a\ogr, =c-/f£
where a, 6, c are constants.

Similarly for other special cases; and, in the most general case, only
a quadrature is necessary.

The tests of functional dependence of one complex on another are of
effective importance in the case when the supposed dependent complex
arises in the form u + iv, where u and v are real; the tests are, of course,
superfluous when w is explicitly given as a function of z. When w does
arise in the form u + iv and satisfies the conditions of functionality, perhaps
the simplest method (other than by inspection) of obtaining the explicit
expression in terms of z is to substitute z — iy for x in u + iv ; the simplified
result must be a function of z alone.

11. Conversely, when w is explicitly given as a function of z and it is
divided into its real and its imaginary parts, these parts individually
satisfy the foregoing conditions attaching to u and v. Thus logr, where r
is the distance of a point z from a point a, is the real part of log (z — a)
and therefore satisfies the equation

11.]

EXAMPLE OF RIEMANN S DEFINITION

13

Again, <f>, the angular coordinate of z relative to the same point a, is
the real part of — i log (z — a) and satisfies the same equation : the more
usual form of <£ being tan"1 {(y — y0)/(® — %o)}> where a = x0 + iy0. Again, if
a point z be distant r from a and r' from b, then log (r/r'\ being the real
part of log {(z — a)l(z — b)\, is a solution of the same equation.

The following example, the result of which will be useful subsequently*, uses the
property that the value of the derivative is independent of the differential element.

z-c

Consider a function

u + iv = w = log

where c' is the inverse of c with regard to a circle centre the origin 0 and radius R.
Then

z-c

* V

:— r>

z-c

and the curves u = constant are circles. Let

W- •
(fig. 4) Oc = r, xOc = a so that c = reat, c'= — eal;

then if

Fig. 4.

the values of X for points in the interior of the circle of radius R vary from zero, when
circle u = constant is the point c, to unity, when the circle u = constant is the circle of
radius R. Let the point K ( = 6eal) be the centre of the circle determined by a value of
X, and let its radius be p ( = %MN}. Then since

cM r ,. cN

we have

whence

r+p-d r d + P~r

— Vp-B Q-p

r r

P =

Now if dn be an element of the normal drawn inwards at z to the circle NzM, we have
dz = dx+idy= — dn . cos ^ - idn . sin ^

--«*<*»,

where ^ ( = zKx'} is the argument of z relative to the centre of the circle. Hence, since

dw 1 1

we have
But

so that

and

, ., , du .dv

and therefore -=- + i -j- =
dn dn

__ _

dz z — c z-c'1
du .dv dw
dn

.dv dw /I 1 \ ty

dn dn \z — c' z — c)

e^ - Reai) •

J> _ 1 _ 1 !_

I /i! ~\ ^^ 7? *1^ X ff */^
\ A7*6 — J\G ./t6 i

* In § 217, in connection with the investigations of Schwarz, by whom the result is stated,
Ges. Werke, t. ii, p. 183.

14 DEFINITIONS [11.

Hence, equating the real parts, it follows that

du (_R2-r2A2)2

dn ~ \R(R*- r2) {E2 - 2Rr\ cos (^ - Q) + XV2} '
the differential element dn being drawn inwards from the circumference of the circle.

The application of this method is evidently effective when the curves u = constant,
arising from a functional expression of w in terms of z, are a family of non-intersecting
algebraical curves.

12. As the tests which are sufficient and necessary to ensure that a
complex quantity is a function of z have been given, we shall assume that
all complex quantities dealt with are functions of the complex variable
(§§ 6, 7). Their characteristic properties, their classification, and some of
the simpler applications will be considered in the succeeding chapters.

Some initial definitions and explanations will now be given.

(i). It has been assumed that the function considered has a differential
coefficient, that is, that the rate of variation of the function in any direction
is independent of that direction by being independent of the mode of change
of the variable. We have already decided (§ 8) not to use the term analytical
for such a function. It is often called monogenic, when it is necessary to
assign a specific name ; but for the most part we shall omit the name, the
property being tacitly assumed*.

We can at once prove from the definition that, when the derivative

/ dw\ •.-.'•, if- c <-• v dw Idw

w, = -p- exists, it is itselt a Junction, .bor w-, =-=— = - =— are equations
\ dz ) dx i dy

which, when satisfied, ensure the existence of w^ ; hence

1 dw-! _ 1 3 (dw\
i dy i dy \d% )
_ d_ (I dw\
dx \i dyj
_dw1
= l)x '

shewing, as in § 8, that the derivative ~ is independent of the direction in

CL2

which dz vanishes. Hence wl is a function of z.

Similarly for all the derivatives in succession.

(ii). Since the functional dependence of a complex is ensured only if the
value of the derivative of that complex be independent of the manner in
which the point z + dz approaches to coincidence with z, a question naturally

* This is in fact done by Biemann, who calls such a dependent complex simply a function.
Weierstrass, however, has proved (§ 85) that the idea of a monogenic function of a complex
variable and the idea of dependence expressible by arithmetical operations are not coextensive.
The definition is thus necessary; but the practice indicated in the text will be adopted, as non-
monogenic functions will be of relatively rare occurrence.

12.] DEFINITIONS 15

suggests itself as to the effect on the character of the function that may be
caused by the manner in which the variable itself has come to the value of z.
If a function have only one value for each given value of the variable,
whatever be the manner in which the variable has come to that value, the
function is called uniform*. Hence two different paths from a point a to a
point z give at z the same value for any uniform function ; and a closed
curve, beginning at any point and completely described by the ^-variable,
will lead to the initial value of w, the corresponding w-curve being closed, if z
have passed through no point which makes w infinite.

The simplest class of uniform functions is constituted by algebraical
rational functions.

(iii). If a function have more than one value for any given value of the
variable, or if its value can be changed by modifying the path in which
the variable reaches that given value, the function is called multiform-]'.
Characteristics of curves, which are graphs of multiform functions corre
sponding to a 2-curve, will hereafter be discussed.

One of the simplest classes of multiform functions is constituted by
algebraical irrational functions.

(iv). A multiform function has a number of different values for the same
value of z, and these values vary with z : the aggregate of the variations of
any one of the values is called a branch of the function. Although the
function is multiform for unrestricted variation of the variable, it often
happens that a branch is uniform when the variable is restricted to
particular regions in the plane.

(v). A point in the plane, at which two or more branches of a multiform
function assume the same value, is called a branch-point^ of the function;
the relations of the branches in the immediate vicinity of a branch-point will
hereafter be discussed.

(vi). A function which is monogenic, uniform and continuous over any
part of the ^-plane is called holomorphic § over that part of the plane. When
•a function is called holomorphic without any limitation, the usual implication
is that the character is preserved over the whole of the plane which is not at
infinity.

The simplest example of a holomorphic function is a rational integral
algebraical polynomial.

* Also monodromic, or monotropic; with German writers the title is eindeutig, occasionally,
einandrig.

t Also polytropic ; with German writers the title is mchrdeittig.

J Also critical point, which, however, is sometimes used to include all special points of a
function ; with German writers the title is Verziveigungspunkt, and sometimes Windungspunkt.
French writers use point de ramification, and Italians punto di giramento and punto di
diramazione.

§ Also synectic.

16 EXAMPLES ILLUSTRATING [12.

(vii). A root (or a zero) of a function is a value of the variable for which
the function vanishes.

The simplest case of occurrence of roots is in a rational integral alge
braical function, various theorems relating to which (e.g., the number of
roots included within a given contour) will be found in treatises on the
theory of equations.

(viii). The infinities of a function are the points at which the value of
the function is infinite. Among them, the simplest are the poles* of the
function, a pole being an infinity such that in its immediate vicinity the
reciprocal of the function is holomorphic.

Infinities other than poles (and also the poles) are called the singular
points of the function : their classification must be deferred until after the
discussion of properties of functions.

(ix). A function which is monogenic, uniform and, except at poles,
continuous, is called a meromorphic function f. The simplest example is a
rational algebraical fraction.

13. The following functions give illustrations of some of the preceding
definitions.

(a) In the case of a meromorphic function

F(z)
111 — — * — -

/<*)'

where F and / are rational algebraical functions without a common factor,
the roots are the roots of F (z) and the poles are the roots of f (z). Moreover,
according as the degree of F is greater or is less than that of f,z = vo is a
pole or a zero of w.

(b) If w be a polynomial of order n, then each simple root of w is a

branch-point and a zero of wm, where m is a positive integer ; z = oo is
a pole of w; and z= oo is a pole but not a branch-point or is an infinity
(though not a pole) and a branch-point of w$ according as n is even or odd.

(c) In the case of the function

1

w-

sn-

z

(the notation being that of Jacobian elliptic functions), the zeros are given by

z
for all positive and negative integral values of m and of m'. If we take

- = iK' + 2mK + Zm'iK' -f £

z

* Also polar discontinuities ; also (§ 32) accidental singularities.

t Sometimes rey-nlar, but this term will be reserved for the description of another property of
functions.

13.] THE DEFINITIONS 17

where £ may be restricted to values that are not large, then

w = (- l)m &sn£

so that, in the neighbourhood of a zero, w behaves like a holomorphic
function. There is evidently a doubly-infinite system of zeros: they are
distinct from one another except at the origin, where an infinite number
practically coincide.

The infinities of w are given by

for all positive and negative integral values of n and of n'. If we take

- = 2nK + Zn'iK' + £

2!

then - = (-l)"sn£

w

so that, in the immediate vicinity of f=0, - is a holomorphic function.

Hence f = 0 is a pole of w. There is thus evidently a doubly-infinite system
of poles ; they are distinct from one another except at the origin, where an
infinite number practically coincide. But the origin is not a pole; the
function, in fact, is there not determinate, for it has an infinite number of
zeros and an infinite number of infinities, and the variations of value are not
necessarily exhausted.

For the function — j , the origin is a point which will hereafter be called

sn-

z

an essential singularity.

F.

CHAPTER II.

INTEGRATION OF UNIFORM FUNCTIONS.

14. THE definition of an integral, that is adopted when the variables
are complex, is the natural generalisation of that definition for real variables
in which it is regarded as the limit of the sum of an infinite number of
infinitesimally small terms. It is as follows : —

Let a and z be any two points in the plane ; and let them be connected
by a curve of specified form, which is to be the path of variation of the
independent variable. Let f(z) denote any function of 0; if any infinity
of f(z) lie in the vicinity of the curve, the line of the curve will be chosen
so as not to pass through that infinity. On the curve, let any number of
points z^ z2,..., zn in succession be taken between a and z ; then, if the sum

(z, - a)f (a) + (z, - z,} f (z,) + ... + (z- zn)f(zn}

have a limit, when n is indefinitely increased so that the infinitely numerous
points are in indefinitely close succession along the whole of the curve from a
to z, that limit is called the integral of / (z) between a and z. It is denoted,
as in the case of real variables, by

f(z)dz.

The limit, as the value of the integral, is associated with a particular
curve : in order that the integral may have a definite value, the curve (called
the path of integration) must, in the first instance, be specified*. The
integral of any function whatever may not be assumed to depend in general
only upon the limits.

15. Some inferences can be made from the definition.
(I.) The integral along any path from a to z passing through a point £ is
the sum of the integrals from a to £ and from \ to z along the same path.

* This specification is tacitly supplied when the variables are real : the variable point moves
along the axis of x.

15.] INTEGRATION 19

Analytically, this is expressed by the equation

P / (*) dz = I V (*) dz + I V (*) <fc,

^ a J a J f

the paths on the right-hand side combining to form the path on the left.

(II.) When the path is described in the reverse direction, the sign of the
integral is changed : that is,

the curve of variation between a and z being the same.

(III.) The integral of the sum of a finite number of terms is equal to
the sum of the integrals of the separate terms, the path of integration being
the same for all.

(IV.) If a function f (z) be finite and continuous along any finite line
between two points a and z, the integral \ f(z)dz is finite.

J a

Let 7 denote the integral, so that we have I as the limit of

r=0

hence |/| = limit of

Because f(z} is finite and continuous, its modulus is finite and therefore
must have a superior limit, say M, for points on the line. Thus

80 that I/I < limit of r+1

<MS,

where 8 is the finite length of the path of integration. Hence the modulus
of the integral is finite ; the integral itself is therefore finite.

No limitation has been assigned to the path, except finiteness in length ;
the proposition is still true when the curve is a closed curve of finite length.

Hermite and Darboux have given an expression for the integral which
leads to the same result. We have as above

f(z)\ dz\,
where 6 is a real positive quantity less than unity. The last integral involves

2—2

20 THEOREMS [15.

only real variables; hence* for some point £ lying between a and z, we have

f

J a

so that l/| = fl9f|/(!)|.

It therefore follows that there is some argument a such that, if X = Be10-,

This form proves the finiteness of the integral ; and the result is the
generalisation f to complex variables of the theorem just quoted for real
variables.

(V.) When a, function is expressed in the form of a series, which converges
uniformly and unconditionally, the integral of the function along any path of
finite length is the sum of the integrals of the terms of the series along the
same path, provided that path lies within the circle of convergence of the series :
— a result, which is an extension of (III.) above.

Let M0 + MI + u.2 + . . . be the converging series ; take

/ (z) = U0 + M! + . . . + Un + R,

where \R\ can be made infinitesimally small with indefinite increase of n,
because the series converges uniformly and unconditionally. Then by (III.),
or immediately from the definition of the integral, we have

rz rs rz rz re

f(z)dz= I u0dz + ^dz + . .. + I undz + 1 Rdz,

J a J a J a J a J a

the path of integration being the same for all the integrals. Hence, if

re n re

(S) = I f (z) dz — 2 I umdz,

J a m=oJ a

ft

we have © = I Rdz.

ft

= I

J a

Let R be the greatest value of \R\ for points in the path of integration
from a to z, and let 8 be the length of this path, so that 8 is finite ;

then, by (IV.),

\®\<SR.

Now 8 is finite ; and, as n is increased indefinitely, the quantity R tends
towards zero as a limit for all points within the circle of convergence and
therefore for all points on the path of integration provided that the path lie
within the circle of convergence. When this proviso is satisfied, |@| becomes
infinitesimally small and therefore also ® becomes infinitesimally small with

* Todhunter's Integral Calculus (4th ed.), § 40; Williamson's Integral Calculus, (Gth ed.), § 96.
t Hermite, Cours d la faculte dcs sciences de Paris (46mc ed., 1891), p. 59, where the reference
to Darboux is given.

15.]

ON INTEGRATION

21

indefinite increase of n. Hence, under the conditions stated in the enuncia
tion, we have

rs oo r%

f(z)dz- 2 I umdz = 0,

J a m^QJ a

which proves the proposition.

16. The following lemma* is of fundamental importance.

Let any region of the plane, on which the ^-variable is represented, be
bounded by one or more simple^ curves which do not meet one another:
each curve that lies entirely in the finite part of the plane will be considered
to be a closed curve.

If ' p and q be any two functions of cc and y, which, for all points within the
region or along its boundary, are uniform, finite and continuous, then the
integral

fffdq dp\j ,
1 1 a - a dxdy,

JJ \dx dyj

extended over the whole area of the region, is equal to the integral

f(pdx + qdy),
taken in a positive direction round the whole boundary of the region.

(As the proof of the proposition does not depend on any special form of
region, we shall take the area to be (fig. 5) that which is included by the
curve QiPiQs'Pa' and excluded by P^Qz'PsQs and excluded by P/P2. The
positive directions of description of the curves are indicated by the arrows ;
and for integration in the area the positive directions are those of increas
ing a; and increasing y.)

AB

Fig. 5.

* It is proved by Eiemann, Ges. Werke, p. 12, and is made by him (as also by Cauchy) the
basis of certain theorems relating to functions of complex variables.

t A curve is called simple, if it have no multiple points. The aim, in constituting the boundary
from such curves is to prevent the superfluous complexity that arises from duplication of area on
the plane. If, in any particular case, multiple points existed, the method of meeting the difficulty
would be to take each simple loop as a boundary.

22 FUNDAMENTAL THEOREM [16.

First, suppose that both p and q are real. Then, integrating with regard
to x, we have *

where the brackets imply that the limits are to be introduced. When the
limits are introduced along a parallel GQ^... to the axis of x, then, since
CQiQi'. • • gives the direction of integration, we have

[qdy] = - qjdyj. + qi'dt/i - q.2dy2 + q-2'dy2' - q3dy3 + q»dy9',

where the various differential elements are the projections on the axis of y
of the various elements of the boundary at points along GQiQJ....

Now when integration is taken in the positive direction round the whole
boundary, the part of / qdy arising from the elements of the boundary at the
points on CQjQ/... is the foregoing sum. For at Q3' it is qa'dy3 because the
positive element dy9, which is equal to CD, is in the positive direction of
boundary integration; at Q3 it is —q3dys because the positive element dy3,
also equal to CD. is in the negative direction of boundary integration ;
at Qz it is q2'dy2', for similar reasons ; at Q.2 it is — q2dya, for similar reasons ;
and so on. Hence

corresponding to parallels through C and D to the axis of x, is equal to
the part of fqdy taken along the boundary in the positive direction for all
the elements of the boundary that lie between those parallels. Then when
we integrate for all the elements CD by forming f[qdy], an equivalent is
given by the aggregate of all the parts of fqdy taken in the positive direction
round the whole boundary ; and therefore

on the suppositions stated in the enunciation.
Again, integrating with regard to y, we have

when the limits are introduced along a parallel RP^P^. . . to the axis of y :
the various differential elements are the projections on the axis of x of the
various elements of the boundary at points along SPjP/....

It is proved, in the same way as before, that the part of - jpdx arising
from the positively-described elements of the boundary at the points on
BP^'... is the foregoing sum. At P3 the part of fpdac is - p3'dx3, because
the positive element dx3, which is equal to AB, is in the negative direction

* It is in this integration, and in the corresponding integration for p, that the properties of
the function q are assumed : any deviation from uniformity, finiteness or continuity within the
region of integration would render necessary some equation different from the one given in
the text.

16.] IN INTEGRATION 23

of boundary integration ; at P3 it is p3dx3, because the positive element
dx3, also equal to AB, is in the positive direction of boundary integration;
and so on for the other terms. Hence

- [pdas],

corresponding to parallels through A and B to the axis of y, is equal to
the part of fpdx taken along the boundary in the positive direction for all
the elements of the boundary that lie between those parallels. Hence
integrating for all the elements AB, we have as before

[[dp j j , j

~ dxdy = — I pax,

JJdy

and therefore II U ?r ) dxdy=f(pdx + qdy).

Secondly, suppose that p and q are complex. When they are resolved
into real and imaginary parts, in the forms p' + ip" and q' + iq" respectively,
then the conditions as to uniformity, finiteness and continuity, which apply to
p and q, apply also to p', q', p", q". Hence

and ~ - - dxdy = j(p"dx + q"dy),

and therefore 1 1 [ 2* _ J9 j dxdy = J(pdx + qdy}

JJ \ox oy/

which proves the proposition.

No restriction on the properties of the functions p and q at points
that lie without the region is imposed by the proposition. They may have
infinities outside, they may cease to be continuous at outside points or they
may have branch-points outside ; but so long as they are finite and continuous
everywhere inside, and in passing from one point to another always acquire
at that other the same value whatever be the path of passage in the region,
that is, so long as they are uniform in the region, the lemma is valid.

17. The following theorem due to Cauchy* can now be proved : _
If a function f(z) be holomorphic throughout any region of the z-plane,
then the integral ff(z) dz, taken round the whole boundary of that region, is zero.
We apply the preceding result by assuming

p=f(z\ q = ip = if(z);

owing to the character of f(z), these suppositions are consistent with the

* For an account of the gradual development of the theory and, in particular, for a
statement of Cauchy's contributions to the theory (with references), see Casorati, Teorica
delle funzioni di variabili complcsse, pp. 64-90, 102-106. The general theory of functions,
as developed by Briot and Bouquet in their treatise Theoric des fonctiom ellipUques, is based
upon Cauchy's method.

24 INTEGRATION OF [17.

conditions under which the lemma is valid. Since p is a function of z, we
have, at every point of the region,

dp _ I dp

das i dy '
and therefore, in the present case,

dq _ . dp _ dp

das doc dy '

There is no discontinuity or infinity of p or q within the region ; hence

the integral being extended over the region. Hence also

!(pdx + qdy) = 0, A^ ^/
when the integral is taken round the whole boundary of the region. But

pdx + qdy = pdx + ipdy
— pdz
=f(z)dz,

and therefore //(X) dz = 0,

the integral being taken round the whole boundary of the region within
which f(z) is holomorphic.

It should be noted that the theorem requires no limitation on the cha
racter of/(^) for points z that are not included in the region.

Some important propositions can be derived by means of the theorem, as
follows.

18. When a function f (z) is holomorphic over any continuous region

rz
of the plane, the integral I f(z)dz is a holomorphic function of 2 provided the

J a

points z and a as well as the whole path of integration lie within that region.

The general definition (§ 14) of an integral is associated with a specified
path of integration. In order to prove that the integral is a holomorphic
function of z, it will be necessary to prove (i) that the integral acquires the
same value in whatever way the point z is attained, that is, that the value is
independent of the path of integration, (ii) that it is finite, (iii) that it
is continuous, and (iv) that it is monogenic.

Let two paths ayz and afiz between a and z be drawn (fig. 6) in the
continuous region of the plane within which f(z) is
holomorphic. The line ayzfia is a contour over the area
of which / (z} is holomorphic ; and therefore ff(z) dz
vanishes when the integral is taken along ayzfta.
Dividing the integral into two parts and implying by
Zy, Zp that the point z has been reached by the paths a"
a<yz, a{3z respectively, we have Fig. 6.

18.] HOLOMOEPHIC FUNCTIONS 25

and therefore */ (z) dz = - f (z) dz

J a J Zg

-?/*/(*)*

J a
Thus the value of the integral is independent of the way in which z has

FZ

acquired its value ; and therefore I f(z) dz is uniform in the region. Denote
it by F(z).

Secondly, f(z) is finite for all points in the region and, after the result
of § 17, we naturally consider only such paths between a and z as are finite in
length, the distance between a and z being finite; hence (§ 15, IV.) the
integral F (z} is finite for all points z in the region.

Thirdly, let z' (= z 4- 82) be a point infinitesimally near to z ; and consider
I f(z) dz. By what has just been proved, the path from a to z' can be taken

J d

aftzz' ; therefore

(*/(*) dz = [/(z) dz + lZf(z) dz

J * J a J z

fz+8z rz rz+Sz

or f(z}dz- \ f(z)dz=\ f(z)dz,

J a J a J z

fz+Sz

80 that F(z + Sz) - F(z) = f(z} dz.

J 2

Now at points in the infinitesimal line from z to z' , the value of the
continuous function f(z) differs only by an infinitesimal quantity from its
value at z ; hence the right-hand side is

where e| is an infinitesimal quantity vanishing with ck It therefore follows
that

is an infinitesimal quantity with a modulus of the same order of small
quantities as \Sz\. Hence F (z) is continuous for points z in the region.
Lastly, we have

and therefore F(z + Sz)-F(z)

82

has a limit when Sz vanishes; and this limit, f(z), is independent of the
way in which 8z vanishes. Hence F (z) has a differential coefficient ; the
integral is monogenic for points z in the region.

26 INTEGRATION OF [18.

Hence F (z), which is equal to

* f(z)d*t

is uniform, finite, continuous and monogenic; it is therefore a holomorphic
function of z.

As in § 16 for the functions p and q, so here for f(z), no restriction is
placed on properties of / (z) at points that do not lie within the region; so
that elsewhere it may have infinities, or discontinuities or branch points.
The properties, essential to secure the validity of the proposition, are
(i) that no infinities or discontinuities lie within the region, and (ii) that the
same value of f(z) is acquired by whatever path in the continuous region
the variable reaches its position z.

COROLLARY. No change is caused in the value of the integral of a
holomorphic function between two points when the path of integration between
the points is deformed in any manner, provided only that, during the defor
mation, no part of the path passes outside the boundary of the region within
which the function is holomorphic.

This result is of importance, because it permits special forms of the path
of integration without affecting the value of the integral.

19. When a function f(z) is holomorphic over a part of the plane
bounded by two simple curves (one lying within the other), equal values of
ff(z) dz are obtained by integrating round each of the curves in a direction,
which — relative to the area enclosed by each — is positive.

The ring-formed portion of the plane (fig. 1, p. 3) which lies between
the two curves being a region over which f(z) is holomorphic, the integral
ff(z) dz taken in the positive sense round the whole of the boundary of
the included portion is zero. The integral consists of two parts : first, that
round the outer boundary the positive sense of which is DEF', and second,
that round the inner boundary the positive sense of which for the portion of
area between ABC and DEF is ACE. Denoting the value of ff(z)dz round
DEF by (DEF), and similarly for the other, we have

(ACB) + (DEF) = 0.

The direction of an integral can be reversed if its sign be changed, so that
(ACB) = - (ABC) ; and therefore

(ABC) = (DEF).

But (ABC) is the integral ff(z)dz taken round ABC, that is, round the
curve in a direction which, relative to the area enclosed by it, is positive.

The proposition is therefore proved.

The remarks made in the preceding case as to the freedom from limitations
on the character of the function outside the portion are valid also in this case.

19.] HOLOMORPHIC FUNCTIONS 27

COROLLARY I. When the integral of a function is taken round the whole
of any simple curve in the plane, no change is caused in its value by continuously
deforming the curve into any other simple curve provided that the function
is holomorphic over the part of the plane in which the deformation is effected.

COROLLARY II. When a function f (z) is holomorphic over a continuous
portion of a plane bounded by any number of simple non-intersecting curves,
all but one of which are external to one another and the remaining one of
which encloses them all, the value of the integral jf(z) dz taken positively round
the single external curve is equal to the sum of the values taken round each of
the other curves in a direction which is positive relative to the area enclosed
by it.

These corollaries are of importance in finding the value of the integral
of a meromorphic function round a curve which encloses one or more of the
poles. The fundamental theorem for such integrals, also due to Cauchy, is
the following.

20. Let f(z) denote a function which is holomorphic over any region in
the z-plane and let a denote any point within that region, which is not a zero

°ff(2); then

., , 1 f/0) ,

f(a) = ^— • *-*-* az>

2vnJ z-a

the integral being taken positively round the whole boundary of the region.

With a as centre and a very small radius p, describe a circle G, which will
be assumed to lie wholly within the region; this assumption is justifiable
because the point a lies within the region. Because f (z) is holomorphic over
the assigned region, the f unction f(z)l(z — a) is holomorphic over the whole of
the region excluded by the small circle C. Hence, by Corollary II. of § 19, we
have

z-a

the notation implying that the integrations are taken round the whole
boundary B and round the circumference of G respectively.

For points on the circle C, let z — a = peei, so that 9 is the variable for
the circumference and its range is from 0 to 2?r ; then we have

dz

z — a

= id6.

Along the circle f(z)=f (a + peei) ; the quantity p is very small and / is
finite and continuous over the whole of the region so that f(a + peei) differs
from /(«) only by a quantity which vanishes with p. Let this difference
be e, which is a continuous small quantity; then |ej is a small quantity
which, for every point on the circumference of C, vanishes with p. Then

28 INTEGRATION OF [20.

" edO.
o

If E denote the value of the integral on the right-hand side, and 77 the
greatest value of the modulus of e along the circle, then, as in § 15,

/•2ir

i E < I e d6

f

Now let the radius of the circle diminish to zero: then 77 also diminishes
to zero and therefore E , necessarily positive, becomes less than any finite
quantity however small, that is, E is itself zero; and thus we have

z — a
which proves the theorem.

This result is the simplest case of the integral of a meromorphic

f(z}
function. The subject of integration is — — , a function which is monogenic

and uniform throughout the region and which, everywhere except at z = a, is
finite and continuous ; moreover, z = a is a pole, because in the immediate

Z ~~~ CL

vicinity of a the reciprocal of the subject of integration, viz. ^-rr > i-B h°l°-
morphic.

The theorem may therefore be expressed as follows :

If g (z) be a meromorphic function, which in the vicinity of a can be

f(z}
expressed in the form J where f(a) is not zero and which at all other

Z — CL

points in a region enclosing a is holomorphic, then

- — . fg (z) dz = limit of (z — a)g (z) when z — a,

the integral being taken round a curve in the region enclosing the point a.

The pole a of the function g (z) is said to be simple, or of the first order,
or of multiplicity unity.

Corollary. The more general case of a meromorphic function with a
finite number of poles can easily be deduced. Let these be a1}..., an each
assumed to be simple ; and let

G (z) = (z- a,) (z - aa). ..(z - an).

20.] MEROMORPHIC FUNCTIONS 29

Let f(z) be a holomorphic function within a region of the 2-plane bounded
by a simple contour enclosing the n points a1} a»,...an, no one of which is a
zero off(z). Then since

f(z) » 1 f(z)

we have j^~( = S „,, . -^-- .

6r (#) r=i Or (ar) z — ar

w ^ f u f/(*),j 3 ! f/(*) ,7

We therefore have "L , ' dz = 2< >.. , . I dz,

J &(*) r=iCr (ar)J 2-ar

each integral being taken round the boundary. But the preceding proposition
gives

because f(z) is holomorphic over the whole region included in the contour ;
and therefore

the integral on the left-hand side being taken in the positive direction*.

The result just obtained expresses the integral of the meromorphic
function round a contour which includes a finite number of its simple poles.
It can be otherwise obtained by means of Corollary II. of § 19, by adopting
a process similar to that adopted above, viz., by making each of the curves in
the Corollary quoted small circles round the points Oj,..., an with ultimately
vanishing radii.

21. The preceding theorems have sufficed to evaluate the integral of
a function with a number of simple poles : we now proceed to obtain
further theorems, which can be used among other purposes to evaluate
the integral of a function with poles of order higher than the first.

We still consider a function f(z) which is holomorphic within a given
region. Then, if a be a point within the region which is not a zero of f(z),
we have

z - a

the point a being neither on the boundary nor within an infinitesimal
distance of it. Let a + Sa be any other point within the region ; then

dz,

z — a — 8a

* We shall for the future assume that, if no direction for a complete integral be specified, the
positive direction is taken.

30

and therefore

PROPERTIES OF

[21.

iff,

8a

f(z)dz,

t J ((* - a)2 (z - of (z-a -Sa)j
the integral being in every case taken round the boundary.

Since f(z) is monogenic, the definition of /'(a), the first derivative of
/(a), gives /'(a) as the limit of

f(a + Ba)-f(a)
Ba

when Ba ultimately vanishes ; hence we may take

where a is a quantity which vanishes with Ba and is therefore such that \ a \
also vanishes with Ba. Hence

dividing out by Sa and transposing, we have

As yet, there is no limitation on the value of Sa ; we now proceed to a limit
by making a + Ba approach to coincidence with a, viz., by making Ba
ultimately vanish. Taking moduli of each of the members of the last
equation, we have

(a) _ i f J(*

2in j (z - o

_„ + ««.

(z — a)2 (z — a — Ba)

27T

dz

Let the greatest modulus of -. ~ =r—. for points z along the

(j — a)2 (z — a — Ba)

boundary be M, which is a finite quantity on account of the conditions
applying to f(z) and the fact that the points a and a + Ba are not
infinitesimally near the boundary. Then, by § 15,

t

dz

'0-a)2 (z-a-Ba)

<MS,

where 8 is the whole length of the boundary, a finite quantity. Hence

1 f f(z} , , |8a|

dz

c

ITT

21.] HOLOMORPHIC FUNCTIONS 31

When we proceed to the limit in which Sa vanishes, we have Ba = 0
and |o-| = 0, ultimately; hence the modulus on the left-hand side ultimately
vanishes and therefore the quantity to which that modulus belongs is itself
zero, that is,

,

(z — of

so that / (a) = —-. !/-^~n dz.

ZTTI )(z- of

This theorem evidently corresponds in complex variables to the well-known
theorem of differentiation with respect to a constant under the integral
sign when all the quantities concerned are real.

Proceeding in the same way, we can prove that

/ (a + &*)-/ (a) _ 2!_ f /(*)
Ba ~2Trij(z-af

where 6 is a small quantity which vanishes with Ba. Moreover the integral
on the right-hand side is finite, for the subject of integration is everywhere
finite along the path of integration which itself is of finite length. Hence,
first, a small change in the independent variable leads to a change of the
same order of small quantities in the value of the function f (a), which
shews that f (a) is a continuous function. Secondly, denoting

&*) -/(a)

by &/'(«), we have the limiting value of -— *— - equal to the integral on

the right-hand side when Sa vanishes, that is, the derivative of f (a) has
a value independent of the form of 8a and therefore /' (a) is monogenic.
Denoting this derivative by /"(a), we have

J (z — a)3

Thirdly, the function f (a) is uniform ; for it is the limit of the value
of — - -- x-- — J-\J and both /(a) and /(a + Sa) are uniform. Lastly, it

is finite; for (S 15) it is the value of the integral - — . l.^—^dz, in which

2?n J (z — af

the length of the path is finite and the subject of integration is finite at
every point of the path.

Hence f (a) is continuous, monogenic, uniform, and finite throughout
the whole of the region in which f (z) has these properties: it is a
holomorphic function. Hence : —

When a function is holomorphic in any region of the plane bounded

32 PROPERTIES OF [21.

by a simple curve, its derivative is also holomorphic within that region. And,
by repeated application of this theorem : —

When a function is holomorphic in any region of the plane bounded
by a simple curve, it has an unlimited number of successive derivatives each
of which is- holomorphic within the region.

All these properties have been shewn to depend simply upon the holo
morphic character of the fundamental function ; but the inferences relating
to the derivatives have been proved only for points within the region and
not for points on the boundary. If the foregoing methods be used to prove
them for points on the boundary, they require that a consecutive point shall
be taken in any direction ; in the absence of knowledge about the fundamental
function for points outside (even though just outside) no inferences can be
justifiably drawn.

An illustration of this statement is furnished by the hypergeometric series
which, together with all its derivatives, is holomorphic within a circle of
radius unity and centre the origin ; and the series converges unconditionally
everywhere on the circumference, provided 7 > a. + /3. But the corresponding
condition for convergence on the circumference ceases to be satisfied for some
one of the derivatives and for all which succeed it : as such functions do not
then converge unconditionally, the circumference of the circle must be
excluded from the region within which the derivatives are holomorphic.

22. Expressions for the first and the second derivatives have been
obtained.

By a process similar to that which gives the value of f (a), the derivative
of order n is obtainable in the form

n ' f f (z\

/<») (a) = — . I, ' dz,
J w 2wt J (z - a)n+l

the integral being taken round the whole boundary of the region or round any
curves which arise from deformation of the boundary, provided that no point
of the curves in the final or any intermediate form is indefinitely near to a.

In the case when the curve of integration is a circle, no point of which
circle may lie outside the boundary of the region, we have a modified form
fcr /*'(•>

For points along the circumference of the circle with centre a and radius

r, let

z — a = reei,

dz

so that as before — = idO :

z — a

then 0 and 2?r being taken as the limits of 0, we have

22.] HOLOMORPHIC FUNCTIONS 33

Let M be the greatest value of the modulus of f (z) for points on the
circumference (or, as it may be convenient to consider, of points on or within
the circumference) : then

\f(n)(a)\<~ e-nei\\f(a

i / \ / 1 ^ 27ryw * -

nl

<

M

Now, let there be a function <£ (s) defined by the equation

M

— a

which can evidently be expanded in a series of ascending powers of z — a
that converges within the circle. The series is

-

[dnd> (z)~\ , M

Hence —!L±J =n\ —

[ d*» ]z=a *>•

so that, if the value of the nth derivative of $(z), when z = a, be denoted
by <£<n> (a), we have

|/»(a)| «p>(a).

These results can be extended to functions of more than one variable :
the proof is similar to the foregoing proof. When the variables are two,
say z and z', the results may be stated as follows : —

^ For all points z within a given simple curve 0 in the ^-plane and all
points / within a given simple curve G' in the /-plane, let / (z, z) be a
holomorphic function; then, if a be any point within C and a' any point
within G',

^n+nJ (a, a')

J (z - a)n+1 (z' — aTf

where n and ri are any integers and the integral is taken positively round the
two curves G and G'.

If M be the greatest value of \f (z, z'} for points z and z within their
respective regions when the curves G and G' are circles of radii r, r' and
centres a, a', then

dn+n'f(a, a') M

~3aW»' <w!/i!rv^5

F.

34 HOLOMORPHIC FUNCTIONS [22-

M
and if $(?>*)

dn+n'f(a,a')

dn+n'(j> (z, z')

then da»da'«

when z = a and z = a' in the derivative of <£ (z, z).

23. All the integrals of meromorphic functions that have been considered
have been taken along complete curves : it is necessary to refer to integrals
along curves which are lines only from one point to another. A single
illustration will suffice at present.

Consider the integral f -t-^-dz; the function /» is

J H0 z — a

supposed holomorphic in the given region, and z and z0 are
any two points in that region. Let some curves joining z
to z0 be drawn as in the figure (fig. 7).

~ , •* 2o

is holomorphic over the whole area en- Fig> 7

z— a

closed by z^zSz0: and therefore we have ^ = 0 when taken round the

boundary of that area. Hence as in the earlier case we have

z — a Jz0 z — a
The point a lies within the area enclosed by z0yz^z0, and the function

is holomorphic, except in the immediate vicinity of z = a ; hence

r f ( v\

I - dz = 2Trif(a),

J z — a

the integral on the left-hand side being taken round Z0yzj3z0. Hence

z — a

Denoting ^-by g(z), the function g (z) has one pole a in the region

£ "~ CL

considered.

The preceding results are connected only with the simplest form of
meromorphic functions; other simple results can be derived by means of the
other theorems proved in §§ 17—21. Those which have been obtained are
sufficient however to shew that : The integral of a meromorphic function
fg(z)dz from one point to another of the region of the function is not in
general a uniform function. The value of the integral is not altered by
any deformation of the path which does not meet or cross a pole of the
function; but the value is altered when the path of integration is so

23.] GENERAL PROPOSITIONS IN INTEGRATION 35

deformed as to pass over one or more poles. Therefore it is necessary to
specify the path of integration when the subject of integration is a mero-
morphic function ; only partial deformations of the path of integration are
possible without modifying the value of the integral.

24. The following additional propositions* are deduced from limiting
cases of integration round complete curves. In the first, the curve becomes
indefinitely small ; in the second, it becomes infinitely large. And in neither,
are the properties of the functions to be integrated limited as in the pre
ceding propositions, so that the results are of wider application.

I. If f(z) be a function which, whatever be its character at a, has no
infinities and no branch-points in the immediate vicinity of a, the value of
ff(z)dz taken round a small circle with its centre at a tends towards zero
when the circle diminishes in magnitude so as ultimately to be merely the
point a, provided that, as z — a diminishes indefinitely, the limit of (z — a)f(z)
tend uniformly to zero.

Along the small circle, initially taken to be of radius r, let

*-a-fl*i *

dz

so that = idO,

z— a

and therefore Sf(z) dz = i\ (z — a)f(z) d6.

Jo

Hence \ff(z)dz\ = I *" (z - a)f(z) d0

Jo

<r\(z-a)f(z)\de
Jo

rzn

< Md0
Jo

where M' is the greatest value of M, the modulus of (z - a)f(z), for points
on the circumference. Since (z - a)f(z) tends uniformly to the limit zero as
| z -a diminishes indefinitely, \jf(z) dz\ is ultimately zero. Hence the integral
itself jf(z)dz is zero, under the assigned conditions.

Note. If the integral be extended over only part of the circumference of
the circle, it is easy to see that, under the conditions of the proposition,
the value offf(z)de still tends towards zero.

COROLLARY. If (z-a)f(z) tend uniformly to a limit k as \z-a\
diminishes indefinitely, the value of ff(z)dz taken round a small circle centre
a tends towards 27rik in the limit.

* The form of the first two propositions, which is adopted here, is due to Jordan, Cours
d' Analyse, t. ii, §§ 285, 286.

3—2

36 GENERAL PROPOSITIONS [24.

Thus the value of [- dz j, taken round a very small circle centre «, where a is

~ d * /2V

not the origin, is zero : the value of f - - - -, round the same circle is -. ( - \ .

J (a — z) (a-M)

Neither the theorem nor the corollary will apply to a function, such as sn —-^

which has the point a for an essential singularity: the value of (z-a)sn^— ^, as
\z-a\ diminishes indefinitely, does not tend (§ 13) to a uniform limit. As a matter of

fact the function sn — has an infinite number of poles in the immediate vicinity of a
z- a

as the limit z—a, is being reached.

II. Whatever be the character of a function f (z} for infinitely large values
ofz, the value ofjf(z) dz, taken round a circle with the origin for centre, tends
towards zero as the circle becomes infinitely large, provided that, as \z\
increases indefinitely, the limit of zf(z) tend uniformly to zero.

Along a circle, centre the origin and radius R, we have z =Eeei, so that

dz .ja
- = idd,

z

r-2ir

and therefore // 0) dz = i zf(z) d6.

Jo

Hence I //(*)<&! = £*

<T zf(z)\dS
Jo

rzn

< Mde

Jo

<

where M' is the greatest value of M, the modulus of zf(z)t for points on
the circumference. When R increases indefinitely, the value of M' is zero
on the hypothesis in the proposition; hence \$f(*)d*\ is ultimately zero.
Therefore the value of ff(z) dz tends towards zero, under the assigned con
ditions.

Note. If the integral be extended along only a portion of the circumfer
ence, the value of jf(z}dz still tends towards zero.

COROLLARY. // zf(z) tend uniformly to a limit k as \z . increases
indefinitely, the value of jf(z) dz, taken round a very large circle, centre the
origin, tends towards %7rik.

Thus the value of J(l -zn}~^dz round an infinitely large circle, centre the origin, is zero
if n > 2, and is 2ir if » = 2.

III. If all the infinities and the branch-points of a function lie in a finite
region of the z-plane, then the value of jf(z) dz round any simple curve, which

24.] IN INTEGRATION 37

includes all those points, is zero, provided the value of zf(z\ as \z\ increases
indefinitely, tends uniformly to zero.

The simple curve can be deformed continuously into the infinite circle
of the preceding proposition, without passing over any infinity or any
branch- point ; hence, if we assume that the function exists all over the plane,
the value of jf(z) dz is, by Cor. I. of § 19, equal to the value of the integral
round the infinite circle, that is, by the preceding proposition, to zero.

Another method of stating the proof of the theorem is to consider
the corresponding simple curve on Neumann's sphere (§ 4). The surface
of the sphere is divided into two portions by the curve*: in one portion lie
all the singularities and the branch-points, and in the other portion there is
no critical point whatever. Hence in this second portion the function is holo-
morphic ; since the area is bounded by the curve we see that, on passing back
to the plane, the excluded area is one over which the function is holomorphic.
Hence, by § 19, the integral round the curve is equal to the integral round
an infinite circle having its centre at the origin and is therefore zero, as
before.

COROLLARY. If, under the same circumstances, the value of zf(z}, as
\z increases indefinitely, tend uniformly to k, then the value of $f(z)dz round
the simple curve is

Thus the value of I — — r along any simple curve which encloses the two points
J (a2 - z2)*

a and - a is 2ir ; the value of

dz

{(!-«") (!-*%•)}*

round any simple curve enclosing the four points 1, -1, T, -7, is zero, k being a non-

1C K

vanishing constant ; and the value of J(l — z2n)~*dz, taken round a circle, centre the origin
and radius greater than unity, is zero when n is an integer greater than 1.

/dz
~ ~ — 771

K*-«i) (*-««)(*-«•)}*

round any circle, which has the origin for centre and includes the three distinct points
€lt e2, e3, is not zero. The subject of integration has 2 = 00 for a branch-point, so that the
condition in the proposition is not satisfied ; and the reason that the result is no longer
valid is that the deformation into an infinite circle, as described in Cor. I. of § 19,
is not possible because the infinite circle would meet the branch-point at infinity.

25. The further consideration of integrals of functions, that do not possess
the character of uniformity over the whole area included by the curve of in
tegration, will be deferred until Chap. ix. Some examples of the theorems
proved in the present chapter will now be given.

* The fact that a single path of integration is the boundary of two portions of the surface
of the sphere, within which the function may have different characteristic properties, will be
used hereafter (§ 104) to obtain a relation between the two integrals that arise according as the
path is deformed within one portion or within the other.

38 EXAMPLES IN [25.

Ex. 1. It is sufficient merely to mention the indefinite integrals (that is, integrals from
an arbitrary point to a point z} of rational, integral, algebraical functions. After the
preceding explanations it is evident that they follow the same laws as integrals of similar
functions of real variables.

/dz
,— ^ , taken round a simple curve.

When n is 0, the value of the integral is zero if the curve do not include the point a,
and it is Ziri if the curve include the point a.

When n is a positive integer, the value of the integral is zero if the curve do not
include the point a (by § 17), and the value of the integral is still zero if the curve do
include the point a (by § 22, for the function f(z) of the text is 1 and all its derivatives
are zero). Hence the value of the integral round any curve, which does not pass through
a, is zero.

We can now at once deduce, by § 20, the result that, if a holomorphic function be
constant along any simple closed curve within its region, it is constant over the whole
area within the curve. For let t be any point within the curve, z any point on it, and C
the constant value of the function for all the points z ; then

B'

mn

2 — t

the integral being taken round the curve, so that

<&M-— t dz

= C
by the above result, since the point t lies within the curve.

Ex. 3. Consider the integral \e~^dz.

In any finite part of the plane, the function e~02 is holomorphic; therefore (§ 17) the
integral round the boundary of a rectangle
(fig. 8), bounded by the lines x= ±a, y = 0,
y=b, is zero : and this boundary can be
extended, provided the deformation remain
in the region where the function is holo
morphic. Now as a tends towards infinity,
the modulus of e~z\ being e~x2 + y2, tends
towards zero when y remains finite ; and
therefore the preceding rectangle can be Fig. 8.

extended towards infinity in the direction of the axis of x, the side b of the rectangle
remaining unaltered.

Along A' A, we have z=x : so that the value of the integral along the part A' A of the

fa

boundary is I e~x dx.

J -a

Along AB, we have z = a + iy, so that the value of the integral along the part AB

f*

is i I e~(a + iyrdy.

Jo
Along BB', we have z = x + ib, so that the value of the integral along the part BB'

f'a
is I e-(x + lVdx.

J a

Along B'A', we have z=-a + iy, so that the value of the integral along the part

B'A1 is i (V(-«H
J t,

25.] INTEGRATION 39

/•ft
The second of these portions of the integral is e~a<i . » . I tP~***tefy, which is easily seen

J o
to be zero when the (real) quantity a is infinite.

Similarly the fourth of these portions is zero.

Hence as the complete integral is zero, we have, on passing to the limit,

I e~^dx+\ e-^2ibx + b'2da;=0,

J -<*> J oo

whence e62 I e~ *-***&?=* I

J -oo J -<*

/oo
e'3^ (cos 2bx—i sin

and therefore, on equating real parts, we obtain the well-known result

/

J -Q

This is only one of numerous examples* in which the theorems in the text can be
applied to obtain the values of definite integrals with real limits and real variables.

rzn-i
Ex. 4. Consider the integral I - --- dz. where n is a real positive quantity less than

J 1+z
unity.

The only infinities of the subject of integration are the origin and the point - 1 ;
the branch-points are the origin and 2=00. Everywhere else in the plane the function
behaves like a holomorphic function ; and, therefore, when we take any simple closed
curve enclosing neither the origin nor the point — 1, the integral of the function round
that curve is zero.

We shall assume that the curve lies on the positive side of the axis of x and that it
is made up of : —

(i) a semicircle (73 (fig. 9), centre the origin and radius R which is made to increase
indefinitely :

Fig. 9.

(ii) two semicircles, ct and c2, with their centres at 0 and — 1 respectively, and with

radii r and /, which ultimately are made infinitesimally small :
(iii) the diameter of (73 along the axis of x excepting those ultimately infinitesimal

portions which are the diameters of cx and of c2.

The subject of integration is uniform within the area thus enclosed although it
is not uniform over the whole plane. We shall take that value of zn~l which has its
argument equal to (n— 1) 6, where 6 is the argument of z.

* See Briot and Bouquet, Theorie des fonctions elliptiques, (2nd ed.), pp. 141 et sqq., from
which examples 3 and 4 are taken.

40 EXAMPLES IN [25.

The integral round the boundary is made up of four parts.

0H — 1

(a) The integral round (73. The value of z . , as z \ increases indefinitely, tends

uniformly to the limit zero ; hence, as the radius of the semicircle is increased indefinitely,
the integral round (73 vanishes (§ 24, n., Note).

^n— 1

(b) The integral round cv The value of z . , as | z \ diminishes indefinitely,

1 -\-z

tends uniformly to the limit zero ; hence as the radius of the semicircle is diminished
indefinitely, the integral round cv vanishes (§ 24, I., Note}.

zn-l

(c) The integral round c2. The value of (1 + 2) , as |1+2| diminishes indefinitely

A ~r z

for points in the area, tends uniformly to the limit (— I)""1, i.e., to the limit g(M~1)'™.
Hence this part of the integral is

being taken in the direction indicated by the arrow round c2) the infinitesimal semicircle.
Evidently -- =id6 and the limits are TT to 0, so that this part of the whole integral is

idd

(d) The integral along the axis of x. The parts at — 1 and at 0 which form the
diameters of the small semicircles are to be omitted ; so that the value is

-l+r' J r
This is what Cauchy calls the principal value* of the integral

/"°° /yH ~ 1

/ •* 7

I dx.

Since the whole integral is zero, we have

ineniri+ I Y — dx = 0.

Let P = I ^ — dx, P' = I dx,

and 0— I dx,

J o 1-a?

principal values being taken in each case. Then, taking account of the arguments, we have

Since iwenvi + P + 1* = 0,

we have P - eH7riQ = - inenni,

* Williamson's Integral Calculus, % 104.

25.] INTEGRATION 41

so that P— Q cos nn — ir sin nn, Q sin ntr = TT cos nir.

Hence I ; dx—P = ir cosec TOTT,

jo 1+a?

dx—Q = ir cot %TT.

. 5. In the same way it may be proved that

.

where n is an integer, a is positive and o> is e*2" .

Jik 6. By considering the integral Je-2^™-1^ round the contour of the sector of a
circle of radius r, bounded by the radii 0=0, 6=a, where a is less than |TT and n is positive,
it may be proved that

„»— 1 „-;

{r';
on proceeding to the limit when r is made infinite. (Briot and Bouquet.)

Ex. 7. Consider the integral I ~~^, where n is an integer. The subject of integration

is meromorphic ; it has for its poles (each of which is simple) the n points o>r for r=0,
1, ..., n-l, where a is a primitive nth root of unity ; and it has no other infinities and no

branch -points. Moreover the value of — — -, as \z\ increases indefinitely, tends uniformly

to the limit zero ; hence (§ 24, in.) the value of the integral, taken round a circle centre
the origin and radius > 1, is zero.

This result can be derived by means of Corollary II. in § 19. Surround each of the
poles with an infinitesimal circle having the pole for centre ; then the integral round
the circle of radius > 1 is equal to the sum of the values of the integral round the
infinitesimal circles. The value round the circle having «r for its centre is, by § 20,

2rri( limit of " , when z = u>r}
\ z - L J

Hence the integral round the large circle

2

n r=n

= 0.

Ex. 8. Hitherto, in all the examples considered, the poles that have occurred have
been simple : but the results proved in § 21 enable us to obtain the integrals of
functions which have multiple poles within an area. As an example, consider the

integral / (1+g2)n + i round any curve which includes the point i but not the point - i, these
points being the two poles of the subject of integration, each of multiplicity n + l.

42 EXAMPLES IN INTEGRATION [25.

We have seen that /" (a) = j^ J ^_a^n + i *»

where /(«) is holomorphic throughout the region bounded by the curve round which the
integral is taken.

In the present case a is i, and f(z) = . «.n ^\ ', s° that

2» ! (-l)n

u. <• /*n - 2-

and therefore /" (*J = ^j (2i)»*-n "~ ~ wT

Hence we have "***™'

In the case of the integral of a function round a simple curve which contains several
of its poles we first (§ 20) resolve the integral into the sum of the integrals round simple
curves each containing only one of the points, and then determine each of the latter
integrals as above.

Another method that is sometimes possible makes use of the expression of the uniform
function in partial fractions. After Ex. 2, we need retain only those fractions which are of

the form — : the integral of such a fraction is ZniA, and the value of the whole integral

z-a

is therefore tor&A. It is thus sufficient to obtain the coefficients of the inverse first powers
which arise when the function is expressed in partial fractions corresponding to each pole.
Such a coefficient A, the coefficient of -j in the expansion of the function, is called by

Z (Jj

Cauchy the residue of the function relative to the point.
For example,

so that the residues relative to the points -1, -o>, -to2 are f, £«, |«2 respectively.
Hence if we take a semicircle, of radius > 1 and centre the origin with its diameter
along the axis of y, so as to lie on the positive side of the axis of y, the area between the
semi-circumference and the diameter includes the two points -« and -«2 ; and therefore

the value of

dz

taken along the semi-circumference and the diameter, is

&*&»+!•?);

i.e., the value is - *ni.

CHAPTER III.

EXPANSION OF FUNCTIONS IN SERIES OF POWERS.

26. WE are now in a position to obtain the two fundamental theorems
relating to the expansion of functions in series of powers of the variable :
they are due to Cauchy and Laurent respectively.

Cauchy 's theorem is as follows*: —

When a function is holomorphic over the area of a circle of centre a, it can
be expanded as a series of positive integral powers of z-a converging for all
points within the circle.

Let z be any point within the circle; describe a concentric circle of
radius r such that

\z-a\ = p <r<R, ^ ^i.

where R is the radius of the given circle. If t
denote a current point on the circumference of the
new circle, we have

dt

t — a z — a
t — a

Fif?. 10.

the integral extending along the whole circumference of radius r. Now

z-a

t-a

z -an+l

z — a

— a

t—a

so that, by § 14 (III.), we have

J_ f
27ri]

f(t)

t-z\t-a

dt.

* Exercices d' 'Analyse et de Physique Mathe'matiqne, t. ii, pp. 50 et seq. ; the memoir was first
made public at Turin in 1832.

44

CAUCHY'S THEOREM ON THE [26.

Now /(«) is holomorphic over the whole area of the circle ; hence, if t be
not actually on the boundary of the region (§§ 21, 22), a condition secured by
the hypothesis r < R, we have

and therefore

(z-a)n (z-a)n+l

"

Let the last term be denoted by L. Since z — a =p and \t-a\ = r,
it is at once evident that \t-z\^r-p. Let M be the greatest value of
|/(0| for points along the circle of radius r ; then M must be finite, owing to
the initial hypothesis relating tof(z). Taking

f — n — TP6i

v W/ ~~ I C*

so that dt = i(t- a) d6,

P«+> t*m de

we have \L\ = -f

i

^P

Jlf

rn (i — p)

Jffl-:C

\r)
Now r was chosen to be greater than p ; hence as n becomes infinitely

large, we have W infinitesimally small. Also If (1 — ?l is finite.
\r/ V f/

Hence as ?i increases indefinitely, the limit of |i|, necessarily not negative,
is infinitesimally small and therefore, in the same case, L tends towards
zero.

It thus appears, exactly as in § 15 (V.), that, when n is made to increase
without limit, the difference between the quantity f(z) and the first n + 1
terms of the series is ultimately zero ; hence the series is a converging series
having f(z) as the limit of the sum, so that

which proves the proposition under the assigned conditions. It is the form
of Taylor's expansion for complex variables.

Note. The series on the right-hand side is frequently denoted by
P(z — a), where P is a general symbol for a converging series of positive
integral powers of z — a: it is also sometimes* denoted by P(z\a). Con-

* Weierstrass, Abh. am der Functionenlehre, p. 1.

26.] EXPANSION OF A FUNCTION 45

formably with this notation, a series of negative integral powers of z — a

would be denoted by P I - — ) ; a series of negative integral powers of z

\z — a/

either by P (-) or by P(^|oo), the latter implying a series proceeding in
\zj

positive integral powers of a quantity which vanishes when z is infinite,
i.e., in positive integral powers of — .

Z

If, however, the circle can be made of infinitely great radius so that the
function f(z) is holomorphic over the finite part of the plane, the equivalent
series is denoted by G(z — a) and it converges over the whole plane.
Conformably with this notation, a series of negative integral powers of z - a

which converges over the whole plane is denoted by G I - - j .

27. The following remarks on the proof and on inferences from it should
be noticed.

(i) In order that — - - may be expanded in the required form, the
t — z

point z must be taken actually within the area of the circle of radius R ;
and therefore the convergence of the series P (z — a) is not established for
points on the circumference.

(ii) The coefficients of the powers of z — a in the series are the
values of the function and its derivatives at the centre of the circle ; and the
character of the derivatives is sufficiently ensured (§ 21) by the holomorphic
character of the function for all points within the region. It therefore
follows that, if a function be holomorphic within a region bounded by a
circle of centre a, its expansion in a series of ascending powers of z — a
converging for all points within the circle depends only upon the values of
the function and its derivatives at the centre.

But instead of having the values of the function and of all its derivatives
at the centre of the circle, it will suffice to have the values of the holomorphic
function itself over any small region at a or along any small line through
a, the region or the line not being infinitesimal. The values of the
derivatives at a can be found in either case ; for /' (b) is the limit of
{f(b + 86) —f(b)}/8b, so that the value of the first derivative can be found
for any point in the region or on the line, as the case may be ; and so for all
the derivatives in succession.

(iii) The form of Maclaurin's series for complex variables is at once
derivable by supposing the centre of the circle at the origin. We then
infer that, if a function be kolomorphic over a circle, centre tJie origin, it can be

46 DARBOUX'S EXPRESSION [27.

represented in the form of a series of ascending, positive, integral powers of the
variable given by

where the coefficients of the various powers of z are the values of the derivatives
of f(z) at the origin, and the series converges for all points within the circle.

Thus, the function ez is holomorphic over the finite part of the plane ;
therefore its expansion is of the form G (z). The function log (1 4- z) has a
singularity at — 1 ; hence within a circle, centre the origin and radius unity,
it can be expanded in the form of an ascending series of positive integral
powers of z, it being convenient to choose that one of the values of the
function which is zero at the origin. Again, tan"1.?2 has singularities at the
four points z4 = — I, which all lie on the circumference; choosing the value at
the origin which is zero there, we have a similar expansion in a series, con
verging for points within the circle.

Similarly for the function (1 +z)n, which has — 1 for a singularity.

(iv) Darboux's method* of derivation of the expansion of f (z) in
positive powers of z — a depends upon the expression, obtained in § 15 (IV.),
for the value of an integral. When applied to the general term

1 Uz-a\n+i s,.. ,,
f(t)dt,

= L say, it gives L = \r fe^J /(f),

where £ is some point on the circumference of the circle of radius r, and X is

2 ~ fl

a complex quantity of modulus not greater than unity. The modulus of ^

b ~~ a

is less than a quantity which is less than unity ; the terms of the series of
moduli are therefore less than the terms of a converging geometric progres
sion, so that they form a converging series; the limit of \L\, and therefore
of L, can, with indefinite increase of n, be made zero and Taylor's expansion
can be derived as before.

00

Ex. 1. Prove that the arithmetic mean of all values of z~ n 2 avzv, for points lying along

v = 0

a circle |z| = r entirely contained in the region of continuity, is an. (Rouche, Gutzmer.)
Prove also that the arithmetic mean of the squares of the moduli of all values of

00

2 avzv, for points lying along a circle z\ = r entirely contained in the region of continuity,

x = 0

is equal to the sum of the squares of the moduli of the terms of the series for a point on
the circle. (Gutzmer.)

00

Ex. 2. Prove that the function 2 anzn*,

M = 0

is finite and continuous, as well as all its derivatives, within and on the boundary of the
circle |0| = 1, provided a < 1. (Fredholm.)

* Liouville, 3dmc Ser., t. ii, (1876), pp. 291—312.

28.]

LAURENT'S EXPANSION OF A FUNCTION

47

28. Laurent's theorem is as follows*: —

A function, which is holomorphic in a part of the plane bounded by two
concentric circles with centre a and finite radii, can be expanded in the form
of a double series of integral powers, positive and negative, of z — a, the series
converging uniformly and unconditionally in the part of the plane between the
circles.

Let z be any point within the region bounded by the two circles of radii
R and R; describe two concentric circles of
radii r arid r' such that

R>r> z-a >r'> R.

Denoting by t and by s current points on the
circumference of the outer and of the inner
circles respectively, and considering the space
which lies between them and includes the point
z, we have, by § 20,

/w-oL

: — Z ZTTlJs — 2"~ Fig. 11.

a negative sign being prefixed to the second integral because the direction
indicated in the figure is the negative direction for the description of the
inner circle regarded as a portion of the boundary.

Now we have

fz — a"
t — a _ z —

t

' — a Iz — a\*
— a \t-aj

z- a.

+ I . — - +
— a.

1 -

z — a

t — a

this expansion being adopted with a view to an infinite converging series,
z — a

because

t — a

is less than unity for all points t; and hence, by § 15,

_ n\n+l

dt.

— z \t — a/

Now each of the integrals, which are the respective coefficients of powers of
z — a, is finite, because the subject of integration is everywhere finite along
the circle of finite radius, by § 15 (IV.). Let the value of

^r* % '•'-'••-

be 2iriur : the quantity ur is not necessarily equal to /'' (a) -r- r I, because no
* Comptes Rendus, t. xvii, (1843), p. 939.

48 LAURENT'S EXPANSION OF [28.

knowledge of the function or of its derivatives is given for a point within
the innermost circle of radius R'. Thus

_L f/2) dt = u0 + (z - a) u1 + (z- a)2 w2+ +(z- a)nun

2w» J t — z

1 [f (t) (z — a\n+1 -,

- z \t — a

The modulus of the last term is less than

M

where p is z-a and If is the greatest value of \f(t)\ for points along the
circle. Because p < r, this quantity diminishes to zero with indefinite in
crease of n ; and therefore the modulus of the expression

v %

becomes indefinitely small with increase of n. The quantity itself therefore
vanishes in the same limiting circumstance ; and hence

1 . [fl&dt = u0 + (z-<i)u1 + ...... +(z-a)mum+ ...... ,

2-7TI J t — Z

so that the first of the integrals is equal to a series of positive powers. This
series converges uniformly and unconditionally within the outer circle, for
the modulus of the (m + l)th term is less than

which is the (m + l)th term of a converging series*.

As in § 27, the equivalence of the integral and the series can be affirmed
only for points which lie within the outermost circle of radius R.

Again, we have

fs - a\n+1

z-a _ s-a fs - a\n (z-a)

s-z z-a \z-a

z — a

this expansion being adopted with a view to an infinite converging series,

because

s — a

z — a

is less than unity. Hence

1 [/s-a\
. If - -
2?rt J \z-aj

-n+1f(s)
J-~-

,

-ds.
z — s

Chrystal, ii, 124.

28.] A FUNCTION IN SERIES 49

The modulus of the last term is less than

M'

P

where M' is the greatest value of \f(s)\ for points along the circle of radius
r'. With indefinite increase of n, this modulus is ultimately zero ; and thus,
by an argument similar to the one which was applied to the former integral,
we have

.. .. - ..

ZTTI J s — z z — a (z — a)2 (z — a)m

where vm denotes the integral f(s — a)m~lf (s) ds taken round the circle.

As in the former case, the series is one which converges uniformly and
unconditionally; and the equivalence of the integral and the series is valid
for points z that lie without the innermost circle of radius R'.

The coefficients of the various negative powers of z — a are of the form

1 f /(*) d( 1 ^
tori] __ 1_ (s-a)'

(s - a)m
a form that suggests values of the derivatives of f (s) at the point given by

- = 0, that is, at infinity. But the outermost circle is of finite radius ;
s-a

and no knowledge of the function at infinity, lying without the circle, is
given, so that the coefficients of the negative powers may not be assumed
to be the values of the derivatives at infinity, just as, in the former case, the
coefficients ur could not be assumed to be the values of the derivatives at the
common centres of the circles.

Combining the expressions obtained for the two integrals, we have
f(z) = u0 + (z — a) u-i + (z — a)2 w2 + ...

+ (z- a)-1 Vl + (z- a)~2 va+ ....

Both parts of the double series converge uniformly and unconditionally for
all points in the region between the two circles, though not necessarily for
points on the boundary of the region. The whole series therefore converges
for all those points : and we infer the theorem as enunciated.

Conformably with the notation (§ 26, note) adopted to represent Taylor's
expansion, a function f(z) of the character required by Laurent's Theorem
can be represented in the form

the series P1 converging within the outer circle and the series P2 converging
without the inner circle ; their sum converges for the ring-space between the
circles.

F. 4

50 LAURENT'S THEOREM [29.

29. The coefficient u0 in the foregoing expansion is

-1- f £9 dt

torijt-a '

the integral being taken round the circle of radius r. We have

dt =ide

t — a
for points on the circle ; and therefore

d0

so that \u0\<!deMt<M',

J ZTT

M' being the greatest value of Mt, the modulus of f(t), for points along the
circle. If M be the greatest value of \f(z}\ for any point in the whole
region in which f(z) is defined, so that M'^.M, then we have

«o 1 < M,

that is, the modulus of the term independent of z — a in the expansion of
f(z) by Laurent's Theorem is less than the greatest value of \f(z) \ at points
in the region in which it is defined.

Again, (z-a)-mf(z) is a double series in positive and negative powers of
z-a, the term independent of z -a being um; hence, by what has just been
proved, um \ is less than p~m M, where p is z - a . But the coefficient um
does not involve z, and we can therefore choose a limit for any point z. The
lowest limit will evidently be given by taking z on the outer circle of radius
R, so that um < MR~m. Similarly for the coefficients vm ; and therefore we
have the result : —

If f(z) be expanded as by Laurent's Theorem in the form

OO 00

u0+ 2 (z-a)mum+ 2 (z-aY^Vm,

m = l m=l

then \um <MR~m, \vm <MR'm,

where M is the greatest value of \f(z) at points within the region in which

f(z) is defined, and R and R' are the radii of the outer and the inner circles

respectively.

30. The following proposition is practically a corollary from Laurent's
Theorem : —

When a function is holomorphic over all the plane which lies outside a
circle of centre a, it can be expanded in the form of a series of negative integral
powers of z — a, the series converging uniformly and unconditionally everywhere
in that part of the plane.

It can be deduced as the limiting case of Laurent's Theorem when the

30.] EXPANSION IN NEGATIVE POWERS 51

radius of the outer circle is made infinite. We then take r infinitely large,
and substitute for t by the relation

t — a = reei,

so that the first integral in the expression (a), p. 47, for/(^) is

1 f2" d0

t — a

Since the function is holomorphic over the whole of the plane which lies
outside the assigned circle, f(t} cannot be infinite at the circle of radius r
when that radius increases indefinitely. If it tend towards a (finite) limit k,
which must be uniform owing to the hypothesis as to the functional character
of f(z\ then, since the limit of (t — z)/(t — a) is unity, the preceding integral
is equal to k.

The second integral in the same expression (a), p. 47, for f(z) is un
altered by the conditions of the present proposition ; hence we have

f(z) = k + (z- a)~l vl + (z- a)-2Vz + ...,

the series converging uniformly and unconditionally without the circle,
though it does not necessarily converge on the circumference.

The series can be represented in the form

1

\z — a/
conformably with the notation of § 26.

Of the three theorems in expansion which have been obtained, Cauchy's
is the most definite, because the coefficients of the powers are explicitly
obtained as values of the function and of its derivatives at an assigned point.
In Laurent's theorem, the coefficients are not evaluated into simple expres
sions ; and in the corollary frofti Laurent's theorem the coefficients are, as is
easily proved, the values of the function and of its derivatives for infinite
values of the variable. The essentially important feature of all the theorems
is the expansibility of the function in series under assigned conditions.

31. It was proved (§21) that, when a function is holomorphic in any
region of the plane bounded by a simple curve, it has an unlimited number
of successive derivatives each of which is holomorphic in the region. Hence,
by the preceding propositions, each such derivative can be expanded in
converging series of integral powers, the series themselves being deducible
by differentiation from the series which represents the function in the region.

In particular, when the region is a finite circle of centre a, within which
f(z) and consequently all the derivatives off(z) are expansible in converging
series of positive integral powers of z — a, the coefficients of the various
powers of z — a are — save as to numerical factors — the values of the

4—2

52 DEFINITION OF DOMAIN [31.

derivatives at the centre of the circle. Hence it appears that, when a function
is holomorphic over the area of a given circle, the values of the function and all
its derivatives at any point z within the circle depend only upon the variable
of the point and upon the values of the function and its derivatives at the
centre.

32. Some of the classes of points in a plane that usually arise in
connection with uniform functions may now be considered.

(i) A point a in the plane may be such that a function of the variable
has a determinate finite value there, always independent of the path by
which the variable reaches a ; the point a, is called an ordinary point* of the
function. The function, supposed continuous in the vicinity of a, is con
tinuous at a : and it is said to behave regularly in the vicinity of an ordinary
point.

Let such an ordinary point a be at a distance d, not infinitesimal, from
the nearest of the singular points (if any) of the function ; and let a circle of
centre a and radius just less than d be drawn. The part of the z-plane lying
within this circle is calledf the domain of a ; and the function, holomorphic
within this circle, is said to behave regularly (or to be regular) in the domain
of a. From the preceding section, we infer that a function and its derivatives
can be expanded in a converging series of positive integral powers of z — a
for all points z in the domain of a, an ordinary point of the function : and
the coefficients in the series are the values of the function and its derivatives
at a.

The property possessed by the series — that it contains only positive
integral powers of z - a— at once gives a test that is both necessary and
sufficient to determine whether a point is an ordinary point. If the point a
be ordinary, the limit of (z - a) f (z} necessarily is zero when z becomes equal
to a. This necessary condition is also sufficient to ensure that the point is
an ordinary point of the function / (z), supposed to be uniform ; for, since
f(z) is holomorphic, the function (z-a)f(z) is also holomorphic and can be
expanded in a series

M0 -f wa (z — d) + w2 (? — a)2 + • • -,

converging in the domain of a. The quantity u0 is zero, being the value
of (z-a)f(z) at a and this vanishes by hypothesis; hence

(z-a)f (z) = (z — a) {MI + u2(z -a) +...},

shewing that / (z) is expressible as a series of positive integral powers of
z— a converging within the domain of a, or, in other words, that/(*) certainly
has a for an ordinary point in consequence of the condition being satisfied.

* Sometimes a regular point.

t The German title is Umgebung, the French is domaine.

32.] ESSENTIAL SINGULARITY 53

(ii) A point a in the plane may be such that a function / (z) of the
variable has a determinate infinite value there, always independent of the
path by which the variable reaches a, the function behaving regularly for

points in the vicinity of a ; then ^—\ nas a determinate zero value there, so

/ (?)

that a is an ordinary point of --r-r . The point a is called a pole (§12) or
an accidental singularity* of the function.

A test, necessary and sufficient to settle whether a point is an accidental
singularity of a function will subsequently (§ 42) be given.

(iii) A point a in the plane may be such that y (2) has not a determinate
value there, either finite or infinite, though the function is regular for all

points in the vicinity of a that are not at merely infinitesimal distances.

i 1
Thus the origin is of this nature for the functions ez, sn - .

Z

Such a point is called-f* an essential singularity of the function. No
hypothesis is postulated as to the character of the function for points
at infinitesimal distances from the essential singularity, while the relation
of the singularity to the function naturally depends upon this character at
points near it. There may thus be various kinds of essential singularities
all included under the foregoing definition ; their classification is effected
through the consideration of the character of the function at points in their
immediate vicinity. (See § 88.)

One sufficient test of discrimination between an accidental singularity
and an essential singularity is furnished by the determinateness of the value
at the point. If the reciprocal of the function have the point for an ordinary
point, the point is an accidental singularity — it is, indeed, a zero for the
reciprocal. But when the point is an essential singularity, the value of the
reciprocal of the function is not determinate there ; and then the reciprocal,
as well as the function, has the point for an essential singularity.

33. It may be remarked at once that there must be at least one
infinite value among the values which a function can assume at an essential
singularity. For if/ (z) cannot be infinite at a, then the limit of (z — a)f (z)
is zero when z = a, no matter what the non-infinite values of f (z) may be,
that is, the limit is a determinate zero. The function (z — a)f(z) is regular
in the vicinity of a : hence by the foregoing test for an ordinary point,
the point a is ordinary and the value of the uniform function f(z) is

* Weierstrass, Abh. aus der Functionenlehre, p. 2, to whom the name is due, calls it ausser-
wesentliche singuldre Stelle ; the term non-essential is suggested by Mr Cathcart, Harnack, p. 148.
t Weierstrass, I.e., calls it wesentliche singulare Stelle.

54 CONTINUATIONS OF A FUNCTION [33.

determinate, contrary to hypothesis. Hence the function must have at least
one infinite value at an essential singularity.

Further, a uniform function must be capable of assuming any value C at
an essential singularity. For an essential singularity of / (z) is also an

essential singularity of / (z) — G and therefore also of .. \_n • The last

function must have at least one infinite value among the values that it can
assume at the point ; and, for this infinite value, we have / (z) — C at the
point, so that/(f) assumes the assigned value C at the essential singularity*.

34. Let f(z) denote the function represented by a series of powers
Pj (z — a), the circle of convergence of which is the domain of the ordinary
point a of the function. The region over which the function / (z) is holo-
morphic may extend beyond the domain of a, although the circumference
bounding that domain is the greatest of centre a that can be drawn within
the region. The region evidently cannot extend beyond the domain of a in
all directions.

Take an ordinary point b in the domain of a. The value at b of the
function /(V) is given by the series Pj (b — a), and the values at b of all its
derivatives are given by the derived series. All these series converge within
the domain of a and they are therefore finite at b ; and their expressions
involve the values at a of the function and its derivatives.

Let the domain of b be formed. The domain of b may be included in
that of a, and then its bounding circle will touch the bounding circle of the
domain of a internally. If the domain of b be not entirely included in that
of a, part of it will lie outside the domain of a ; but it cannot include the
whole of the domain of a unless its bounding circumference touch that of the
domain of a externally, for otherwise it would extend beyond a in all
directions, a result inconsistent with the construction of the domain of a.
Hence there must be points excluded from the domain of a which are also
excluded from the domain of b.

For all points z in the domain of b, the function can be represented by a
series, say P2 (2 — b), the coefficients of which are the values at b of the
function and its derivatives. Since these values are partially dependent
upon the corresponding values at a, the series representing the function may
be denoted by P2 (z — b, a).

At a point z in the domain of b lying also in the domain of a, the two
series Pl (z — a) and P2 (z — b, a) must furnish the same value for the
function / (V) ; and therefore no new value is derived from the new series P2

* Weierstrass, I.e., pp. 50—52; Durege, Elemente der Theorie der Funktionen, p. 119; Holder,
Math. Ann., t. xx, (1882), pp. 138 — 143 ; Picard, " Memoire sur les fonctions entieres," Annahs de
VEcole Norm. Sup., 2me Ser., t. ix, (1880), pp. 145 — 166, which, in this regard, should be consulted
in connection with the developments in Chapter V. See also § 62.

34.] OVER ITS REGION OF CONTINUITY 55

which cannot be derived from the old series Pj. For all such points the new
series is of no advantage ; and hence, if the domain of b be included in that
of a, the construction of the series P2 (z — b, a) is superfluous. Hence in
choosing the ordinary point b in the domain of a we choose a point, if
possible, that will not have its domain included in that of a.

At a point z in the domain of b, which does not lie in the domain of a,
the series P2 (z — b, a) gives a value for f(z) which cannot be given by
Pl (z — a). The new series P2 then gives an additional representation of the
function ; it is called* a continuation of the series which represents the function
in the domain of a. The derivatives of P2 give the values of f(z) for points
in the domain of b.

It thus appears that, if the whole of the domain of b be not included in
that of a, the function can, by the series which is valid over the whole
of the new domain, be continued into that part of the new domain excluded
from the domain of a.

Now take a point c within the region occupied by the combined domains
of a and b ; and construct the domain of c. In the new domain, the function
can be represented by a new series, say P3(z — c), or, since the coefficients
(being the values at c of the function and of its derivatives) involve the
values at a and possibly also the values at b of the function and of its
derivatives, the series representing the function may be denoted by
Pz(z — c, a, b). Unless the domain of c include points, which are not
included in the combined domains of a and b, the series P3 does not give
a value of the function which cannot be given by Pj or P2: we therefore
choose c, if possible, so that its domain will include points not included in
the earlier domains. At such points z in the domain of c as are excluded
from the combined domains of a and 6, the series P3 (z — c, a, b) gives a value
for f(z) which cannot be derived from P1 or P2 ; and thus the new series
is a continuation of the earlier series.

Proceeding in this manner by taking successive points and constructing
their domains, we can reach all parts of the plane connected with one
another where the function preserves its holomorphic character; their
combined aggregate is called -f the region of continuity of the function.
With each domain, constructed so as to include some portion of the region of
continuity not included in the earlier domains, a series is associated, which is
a continuation of the earlier series and, as such, gives a value of the function
not deducible from those earlier series ; and all the associated series are
ultimately derived from the first.

* Biermann, Theorie der analytischen Functional, p. 170, which may be consulted in
connection with the whole of § 34; the German word is Fortsetzung.
t Weierstrass, I.e., p. 1.

56 DEFINITION OF ANALYTIC FUNCTION [34.

Each of the continuations is called an Element of the function. The
aggregate of all the distinct elements is called a monogenic analytic function :
it is evidently the complete analytical expression of the function in its region
of continuity.

Let z be any point in the region of continuity, not necessarily in the
circle of convergence of the initial element of the function; a value of the
function at z can be obtained through the continuations of that initial
element. In the formation of each new domain (and therefore of each new
element) a certain amount of arbitrary choice is possible ; and there may,
moreover, be different sets of domains which, taken together in a set, each
lead to z from the initial point. When the analytic function is uniform, as
before defined (§ 12), the same value at z for the function is obtained,
whatever be the set of domains. If there be two sets of elements, differently
obtained, which give at z different values for the function, then the ana
lytic function is multiform, as before defined (§ 12) ; but not every change
in a set of elements leads to a change in the value at z of a multiform
function, and the analytic function is uniform within such a region of the
plane as admits only equivalent changes of elements.

The whole process is reversible when the function is uniform. We can
pass back from any point to any earlier point by the use, if necessary, of
intermediate points. Thus, if the point a in the foregoing explanation
be not included in the domain of b (there supposed to contribute a continu
ation of the first series), an intermediate point on a line, drawn in the
region of continuity so as to join a and b, would be taken ; and so on,
until a domain is formed which does include a. The continuation, associated
with this domain, must give at a the proper value for the function and its
derivatives, and therefore for the domain of a the original series Pl(z — a)
will be obtained, that is, Pj (z — a) can be deduced from P2 (z — b, a) the
series in the domain of b. This result is general, so that any one of the
continuations of a uniform function, represented by a power-series, can be
derived from any other; and therefore the expression of such a function in
its region of continuity is potentially given by one element, for all the
distinct elements can be derived from any one element.

35. It has been assumed that the property, characteristic of some of the
functions adduced as examples, of possessing either accidental or essential
singularities, is characteristic of all functions ; it will be proved (§ 40) to hold
for every uniform function which is not a mere constant.

The singularities limit the region of continuity ; for each of the separate
domains is, from its construction, limited by the nearest singularity, and the
combined aggregate of the domains constitutes the region of continuity when

35.]

SCHWARZ S CONTINUATION

57

they form a continuous space*. Hence the complete boundary of the region
of continuity is the aggregate of the singularities of the function-}-.

It may happen that a function has no singularity except at infinity ; the
region of continuity then extends over the whole finite part of the plane but
it does not include the point at infinity.

It follows from the foregoing explanations that, in order to know a
uniform analytic function, it is necessary to know some element of the
function, which has been shewn to be potentially sufficient for the derivation
of the full expression of the function and for the construction of its region of
continuity.

36. The method of continuation of a function, which has just been
described, is quite general ; there is one particular continuation, which is
important in investigations on conformal representations. It is contained in
the following proposition, due to SchwarzJ : —

If an analytic function w of z be defined only for a region 8' in the
positive half of the z-plane and if continuous real values of w correspond to
continuous real values of z, then w can be continued across the axis of real
quantities.

Consider a region 8", symmetrical with S' relative to the axis of real
quantities (fig. 12). Then a function is
defined for the region S" by associating
a value w0, the conjugate of w, with z0,
the conjugate of z.

Let the two regions be combined along
the portion of the axis of ac which is their
common boundary ; they then form a
single region S' + S".

Consider the integrals

Fig. 12.

1 [ w j A ! [ wo
o — • I — i-dz and ^ — -. / —
fcp/fjr-f 2w»./,r«t-

taken round the boundaries of 8' and of 8" respectively. Since w is

* Cases occur in which the region of continuity of a function is composed of isolated spaces,
each continuous in itself, but not continuous into one another. The consideration of such cases
will be dealt with briefly hereafter, and they are assumed excluded for the present : meanwhile,
it is sufficient to note that each continuous space could be derived from an element belonging to
some domain of that space and that a new element would be needed for a new space.

t See Weierstrass, I.e., pp. 1—3 ; Mittag-Leffler, " Sur la representation analytique des fonctions
monogenes uniformes d'une variable independante," Acta Math., t. iv, (1884), pp. 1 et seq.,
especially pp. 1 — 8.

£ Crelle, t. Ixx, (1869), pp. 106, 107, and Ges. Math. Abh., t. ii, pp. 66—68. See also Darboux,
Theorie generate des surfaces, t. i, § 130.

58 SCHWARZ'S CONTINUATION [36.

continuous over the whole area of 8' as well as along its boundary and
likewise w0 relative to 8", it follows that, if the point f be in 8', the value of
the first integral is w (f ) and that of the second is zero ; while, if £ lie in 8",
the value of the first integral is zero and that of the second is w0 (£). Hence
the sum of the two integrals represents a unique function of a point in either
8' or 8". But the value of the first integral is

M wdz J^ [B w Q) dap

I ("' ~~ (f • C\ ' I V>

J ji 2— £ ZTriJ A x — L,

the first being taken along the curve EC. A and the second along the axis
AxB ; and the value of the second integral is

1 CAw0(x)dx 1_ f * W0dz0
2-Tri J B x — £ ZTTI J A *o — £ '
the first being taken along the axis Ex A and the second along the curve

ADB. But

w0 (ac) = w (x),

because conjugate values w and w0 correspond to conjugate values of the
argument by definition of W0 and because w (and therefore also w0) is real
and continuous when the argument is real and continuous. Hence when the
sum of the four integrals is taken, the two integrals corresponding to the
two descriptions of the axis of x cancel and we have as the sum

wdz 1

A

and this sum represents a unique function of a point in 8' + 8". These two
integrals, taken together, are

_L [w'dz
2Tn]z-t'

taken round the whole contour of 8' + 8", where w' is equal to w (f) in the
positive half of the plane and to w0 (^) in the negative half.

For all points £ in the whole region 8' + 8", this integral represents a
single uniform, finite, continuous function of f; its value is w (£) in the
positive half of the plane and is w0 (f) in the negative half; and therefore
w0 (£) is the continuation into the negative half of the plane of the function,
which is defined by w (£) for the positive half.

For a point c on the axis of x, we have

w (z) -w(c) = A(z-c) + B(z-cy>+C(z-cY + ...;

and all the coefficients A, B, C,... are real. If, in addition, w be such a
function of z that the inverse functional relation makes z a uniform
analytic function of w, it is easy to see that A must not vanish, so that the
functional relation may be expressed in the form

w(z)—w (c) = (z-c}P(z- c),
where P (z — c) does not vanish when z = c.

CHAPTER IV.

GENERAL PROPERTIES OF UNIFORM FUNCTIONS, PARTICULARLY OF THOSE
WITHOUT ESSENTIAL SINGULARITIES.

37. IN the derivation of the general properties of functions, which will be
deduced in the present and the next three chapters from the results already
obtained, it is to be supposed, in the absence of any express statement to
other effect, that the functions are uniform, monogenic and, except at either
accidental or essential singularities, continuous*.

THEOREM I. A function, which is constant throughout any region of the
plane not infinitesimal in area, or which is constant along any line not infini
tesimal in length, is constant throughout its region of continuity.

For the first part of the theorem, we take any point a in the region of the
plane where the function is constant, and we draw a circle of centre a and
of any radius, provided only that the circle remains within the region of
continuity of the function. At any point z within this circle we have

/<*) =/(a) + (z - a)f (a) + (-, ~^ f" (a) + . . ,

a converging series the coefficients of which are the values of the function
and its derivatives at a. But

/X«) = Limit of ^±MZ/^), :. V, ' :•

which is zero because f(a + Ba) is the same constant as f(a) : so that the
first derivative is zero at a. Similarly, all the derivatives can be shewn to
be zero at a ; hence the above series after its first term is evanescent,
and we have

/(*)-/<«),

that is, the function preserves its constant value throughout its region of
continuity.

The second result follows in the same way, -when once the derivatives are
proved zero. Since the function is monogenic, the value of the first and

* It will be assumed, as in § 35 (note, p. 57), that the region of continuity consists of a single
space ; functions, with regions of continuity consisting of a number of separated spaces, will be
discussed in Chap. VII.

60 ZEROS OF A [37.

of each of the successive derivatives will be obtained, if we make the
differential element of the independent variable vanish along the line.

Now, if a be a point on the line and a + 8a a consecutive point, we have
f(a + So) = f(a) ; hence /' (a) is zero. Similarly the first derivative at any
other point on the line is zero. Therefore we have /' (a + So) =f (a), for
each has just been proved to be zero : hence /" (a) is zero ; and similarly the
value of the second derivative at any other point on the line is zero. So on
for all the derivatives : the value of each of them at a is zero.

Using the same expansion as before and inserting again the zero values
of all the derivatives at a, we find that

/(*)=/(«),

so that under the assigned condition the function preserves its constant value
throughout its region of continuity.

It should be noted that, if in the first case the area be so infinitesimally
small and in the second the line be so infinitesimally short that consecutive
points cannot be taken, then the values at a of the derivatives cannot be
proved to be zero and the theorem cannot then be inferred.

COROLLARY I. If two functions have the same value over any area of
their common region of continuity which is not infinitesimally small or along
any line in that region which is not infinitesimally short, then they have the
same values at all points in their common region of continuity.

This is at once evident : for their difference is zero over that area or along
that line and therefore, by the preceding theorem, their difference has a
constant zero value, that is, the functions have the same values, everywhere
in their common region of continuity.

But two functions can have the same values at a succession of isolated
points, without having the same values everywhere in their common region
of continuity ; in such a case the theorem does not apply, the reason being
that the fundamental condition of equality over a continuous area or along
a continuous line is not satisfied.

COROLLARY II. A function cannot be zero over any continuous area of its
region of continuity which is not infinitesimal or along any line in that region
which is not infinitesimally short without being zero everywhere in its region of
continuity.

This corollary is deduced in the same manner as that which precedes.

If, then, there be a function which is evidently not zero everywhere, we
conclude that its zeros are isolated points though such points may be multiple
zeros.

Further, in any finite area of the region of continuity of a function that is
subject to variation, there can be at most only a finite number of its zeros, when

37.] UNIFORM FUNCTION 61

no point of the boundary of the area is infinitesimally near an essential
singularity. For if there were an infinite number of such points in any
such region, there must be a cluster in at least one area or a succession
along at least one line, infinite in number and so close as to constitute a
continuous area or a continuous line where the function is everywhere zero.
This would require that the function should be zero everywhere in its region
of continuity, a condition excluded by the hypothesis.

And it immediately follows that the points (other than those infini
tesimally near an essential singularity) in a region of continuity, at which a
function assumes any the same value, are isolated points ; and that only a
finite number of such points occur in any finite area.

38. THEOREM II. The multiplicity m of any zero a of a function is
finite provided the zero be an ordinary point of the function, which is not zero
throughout its region of continuity; and the function can be expressed in the

where <f> (z) is holomorphic in the vicinity of a, and a is not a zero of <£ (z).

Let f(z) denote the function ; since a is a zero, we have f(a) = 0.
Suppose that /'(a), f" (a), ...... vanish: in the succession of the derivatives

of f, one of finite order must be reached which does not have a zero value.
Otherwise, if all vanish, then the function and all its derivatives vanish at a;
the expansion of f(z) in powers of z — a leads to zero as the value of f (z\
that is, the function is everywhere zero in the region of continuity, if all the
derivatives vanish at a.

Let, then, the wth derivative be the first in the natural succession which
does not vanish at a, so that m is finite. Using Cauchy's expansion, we have

(? — n\tm) ( ~ _ n\(m+\)

f(z) = (Z a /« (a) + S£Za_/F* (a) + . . .

J m ! J (m + 1) ! J

= (z-ay*$(z\

where <£ (z) is a function that does not vanish with a and, being the quotient
of a converging series by a monomial factor, is holomorphic in the immediate
vicinity of a.

COROLLARY I. If infinity be a zero of a function of multiplicity m and
at the same time be an ordinary point of the function, then the function can be

expressed in the form z~m $ f-J ,

where </>(-) is a function that is continuous and non-evanescent for infinitely

large values of z.

The result can be derived from the expansion in § 30 in the same way as
the foregoing theorem from Cauchy's expansion.

62 ZEROS OF A [38.

COROLLARY II. The number of zeros of a function, account being taken of
their multiplicity, which occur within a finite area of the region of continuity
of the function, is finite, when no point of the boundary of the area is infinitesi-
mally near an essential singularity.

By Corollary II. of § 37, the number of distinct zeros in the limited area
is finite, and, by the foregoing theorem, the multiplicity of each is finite ;
hence, when account is taken of their respective multiplicities, the total
number of zeros is still finite.

The result is, of course, a known result for an algebraical polynomial ; but
the functions in the enunciation are not restricted to be of the type of
algebraical polynomials.

Note. It is important to notice, both for the Theorem and for Corollary I,
that the zero is an ordinary point of the function under consideration ; the
implication therefore is that the zero is a definite zero and that in the
immediate vicinity of the point the function can be represented in the form

P(z — a) or P [-] , the function P(a — a) or P (— ) being .always a definite

\<6 / \ /

zero.

Instances do occur for which this condition is not satisfied. The point
may not be an ordinary point, and the zero value may be an indeterminate
zero ; or zero may be only one of a set of distinct values though everywhere
in the vicinity the function is regular. Thus the analysis of § 13 shews that

z=a is a point where the function sn - - has any number of zero values and

Z CL

any number of infinite values, and there is no indication that there are not
also other values at the point. In such a case the preceding proposition does
not apply ; there may be no limit to the order of multiplicity of the zero, and
we certainly cannot infer that any finite integer m can be obtained such that

(z - a)~m <j> (z)

is finite at the point. Such a point is (§ 32) an essential singularity of the
function.

39. THEOREM III. A multiple zero of a function is a zero of its
derivative ; and the multiplicity for the derivative is less or is greater by
unity according as the zero is not or is at infinity.

If a be a point in the finite part of the plane which is a zero of f(z)
of multiplicity n, we have

/(f)-(*T.a)» + («X

and therefore /' (z) = (z - a)n~l [n$ (z} + (z-a) $ (z)}.
The coefficient of (z — a)n~l is holomorphic in the immediate vicinity of a and
does not vanish for a ; hence a is a zero for /' (z) of decreased multiplicity

39.] UNIFORM FUNCTION

If z = oo be a zero off(z) of multiplicity r, then

where <£ (-) is holomorphic for very large values of z and does not vanish at

\z /

infinity. Therefore

The coefficient of ^~r~1 is holomorphic for very large values of z, and does
not vanish at infinity ; hence z=<x> is a zero off (z) of increased multiplicity
r + l.

Corollary I. If a function be finite at infinity, then z = oo is a zero of the
first derivative of multiplicity at least two.

Corollary II. If a be a finite zero off(z) of multiplicity n, we have
f(z)= n #(z)
f(z) ir-** fW

Now a is not a zero of <J> (z) ; and therefore ^4^r is finite, continuous, uniform

9W

and monogenic in the immediate vicinity of a. Hence, taking the integral
of both members of the equation round a circle of centre a and of radius
so small as to include no infinity and no zero, other than a, of / (z) _ and
therefore no zero of $(z) — we have, by § 17 and Ex. 2, § 25,

~jT/ \ ^"^ ~ ^-

/(*)

40. THEOREM IV. A function must have an infinite value for some finite
or infinite value of the variable.

If M be a finite maximum value of the modulus for points in the plane,
then (§ 22) we have

where r is the radius of an arbitrary circle of centre a, provided the whole of
the circle is in the region of continuity of the function. But as the function
is uniform, monogenic, finite and continuous everywhere, this radius can be
increased indefinitely ; when this increase takes place, the limit of

is zero and therefore /<»> (a) vanishes. This is true for all the indices 1,2,...
of the derivatives.

64 INFINITIES OF A [40.

Now the function can be represented at any point z in the vicinity of a
by the series

which degenerates, under the present hypothesis, to /(a), so that the function
is everywhere constant. Hence, if a function has not an infinity somewhere
in the plane, it must be a constant.

The given function is not a constant; and therefore there is no finite
limit to the maximum value of its modulus, that is, the function acquires
an infinite value somewhere in the plane.

COROLLARY I. A function must have a zero value for some finite or
infinite value of the variable.

For the reciprocal of a uniform monogenic analytic function is itself a
uniform monogenic analytic function ; and the foregoing proposition shews
that this reciprocal must have an infinite value for some value of the
variable, which therefore is a zero of the function.

COROLLARY II. A function must assume any assigned value at least once.

COROLLARY III. Every function which is not a mere constant must have
at least one singularity, either accidental or essential. For it must have
an infinite value : if this be a determinate infinity, the point is an accidental
singularity (§ 32) ; if it be an infinity among a set of values at the point, the
point is an essential singularity (§§ 32, 33).

41. Among the infinities of a function, the simplest class is that con
stituted by its accidental singularities, already defined (§ 32) by the property
that, in the immediate vicinity of such a point, the reciprocal of the function
is regular, the point being an ordinary (zero) point for that reciprocal.

THEOREM V. A function, which has a point cfor an accidental singularity,
can be expressed in the foi*m

(z - c}~n (f> (z),

where n is a finite positive integer and <f> (z) is a continuous function in the
vicinity of c.

Since c is an accidental singularity of the function f(z}, the function ^y-r

/ (z)

is regular in the vicinity of c and is zero there (§ 32). Hence, by § 38, there
is a finite limit to the multiplicity of the zero, say n (which is a positive
integer), and we have

where ^ (z) is uniform, monogenic and continuous in the vicinity of c and is
not zero there. The reciprocal of ^ (z), say <f> (z), is also uniform, monogenic

41.] UNIFORM FUNCTION 65

and continuous in the vicinity of c, which is an ordinary point for (f> (z) ;
hence we have

f(z} = (Z-c)-^(z\

which proves the theorem.

The finite positive integer n measures the multiplicity of the accidental
singularity at c, which is sometimes said to be of multiplicity n or of
order n.

Another analytical expression for f(z) can be derived from that which
has just been obtained. Since c is an ordinary point for <f> (z) and not a zero,
this function can be expanded in a series of ascending, positive, integral
powers of z — c, converging in the vicinity of c, in the form

£(*) = P(*-c)

= uQ + ul(z-c} + ... + un^(z-c)n-l+un(z-c)n+...

= u0 + u,(z - c) + ... + un_^(z - c)71-1 + (z- c)nQ(z-c),
where Q(z — c), a series of positive, integral, powers of z — c converging in the
vicinity of c, is a monogenic analytic function of z. Hence we have

^ = ^» + (7^+ - +,~; + «('-')>

the indicated expression for f(z), valid in the immediate vicinity of c, where
Q (z — c) is uniform, finite, continuous and monogenic.

COROLLARY. A function, which has z= oo for an accidental singularity of
multiplicity n, can be expressed in the form

_

where </>(-) is a continuous function for very large values of \z , and is not

\zj

zero when z = oo . It can also be expressed in the form

1 + ... + an^ z + Q (-} ,
\zj

where Q ( - j is uniform, finite, continuous and monogenic for very large values

f\»\.

The derivation of the form of the function in the vicinity of an accidental
singularity has been made to depend upon the form of the reciprocal of the
function. Whatever be the (finite) order of that point as a zero of the
reciprocal, it is assumed that other zeros of the reciprocal are not at merely
infinitesimal distances from the point, that is, that other infinities of the
function are not at merely infinitesimal distances from the point.

Hence the accidental singularities of a function are isolated points ; and
there is only a finite number of them in any limited portion of the plane.
F. 5

66 INFINITIES OF A [42.

42. We can deduce a criterion which determines whether a given singu
larity of a function /(f) is accidental or essential.

When the point is in the finite part of the plane, say at c, and a finite
positive integer n can be found such that

is not infinite at c, then c is an accidental singularity.

When the point is at infinity and a finite positive integer n can be found
such that

is not infinite when z = oc , then z = oo is an accidental singularity.

If one of these conditions be not satisfied, the singularity at the point is
essential. But it must not be assumed that the failure of the limitation to
finiteness in the multiplicity of the accidental singularity is the only source
or the complete cause of essential singularity.

Since the association of a single factor with the function is effective in
preventing an infinite value at the point when one of the conditions is
satisfied, it is justifiable to regard the discontinuity of the function at
the point as not essential and to call the singularity either non-essential
or accidental (§ 82).

43. THEOREM VI. The poles of a function, that lie in the finite part
of the plane, are all the poles (of increased multiplicity) of the derivatives of
the function that lie in the finite part of the plane.

Let c be a pole of the function f(z) of multiplicity p : then, for any point
z in the vicinity of c,

where </> (z) is holomorphic in the vicinity of c, and does not vanish for z = c.
Then we have

f'(2) = (z~ c)~p $' (z) ~ P (2 ~ c) p 1 $ W
= (z-c)-P-*{(z-c)<j>'(z)-p<}>(z)}

where % (z) is holomorphic in the vicinity of c, and does not vanish for z = c.

Hence c is a pole of/' (z) of multiplicity ^9 + 1. Similarly it can be shewn
to be a pole of /(r) (z) of multiplicity p + r.

This proves that all the poles of f(z) in the finite part of the plane are
poles of its derivatives. It remains to prove that a derivative cannot have
a pole which the original function does not also possess.

Let a be a pole off'(z) of multiplicity m : then, in the vicinity of a,f'(z)
can be expressed in the form

43.] UNIFORM FUNCTION £7

where ^ (z) is holomorphic in the vicinity of a and does not vanish for z = a
Thus

and therefore f (*) = - . + j_ ,

y v ' JlV^ •<*-«)*"*

so that, integrating, we have

f(z}= *(«) _*>)

m 0 - a)™-1 (m - 1) 0 - a)™-2
that is, a is a pole of/0).

An apparent exception occurs in the case when m is unity: for then
we have

the integral of which leads to

f(z} = ^ (a) log (z - a) + . . . ,

so that/0) is no longer uniform, contrary to hypothesis. Hence a derivative
cannot have a simple pole in the finite part of the plane ; and so the exception
is excluded.

The theorem is thus proved.

COROLLARY I. The rth derivative of a function cannot have a pole in the
finite part of the plane of multiplicity less than r + 1.

COROLLARY II. If c be a pole of f (z) of any order of multiplicity ^ and
if f(r] (z) be expressed in the form

, _ Oi__

» / _. _\., _!_*• _ 1 I ••••••*

(Z - CY+T (Z-

there are no terms in this expression with the indices - 1, - 2, ...... , - r.

COROLLARY III. If c be a pole of/ (z) of multiplicity p, we have

=
f(z) z-c~* 4>(z)'

where $ (z) is a holomorphic function that does not vanish for z = c, so that

<£' 0) •

-T-/JN is a holomorphic function in the vicinity of c. Taking the integral of

f'(z)

-j-j~\ round a circle, with c for centre, with radius so small as to exclude all

other poles or zeros of the function f (z), we have

5—2

(}8 INFINITIES OF A [43.

COROLLARY IV. If a simple closed curve include a number N of zeros of
a uniform function f (z) and a number P of its poles, in both of which
numbers account is taken of possible multiplicity, and if the curve contain
no essential singularity of the function, then

the integral being taken round the curve.

f (z)
The only infinities of the function ' ^i within the curve are the zeros

j(z)

and the poles of / (z). Round each of these draw a circle of radius so small
as to include it but no other infinity ; then, by Cor. II. § 18, the integral
round the closed curve is the sum of the values when taken round these
circles. By the Corollary II. § 39 and by the preceding Corollary III., the
sum of these values is

= 2w — %>

= N-P.

It is easy to infer the known theorem that the number of roots of an
algebraical polynomial of order n is n, as well as the further result that
2^ (N - P) is the variation of the argument of / (z) as z describes the
closed curve in a positive sense.

Ex. Prove that, if F(z) be holomorphic over an area, of simple contour, which con
tains roots «!, «2,... of multiplicity m» m2,... and poles cx, c2)... of multiplicity p^ p2J...
respectively of a function f(z) which has no other singularities within the contour, then

the integral being taken round the contour.

In particular, if the contour contains a single simple root a and no singularity, then that
root is given by

the integral being taken as before. (Laurent.)

44. THEOREM VII. If infinity be a pole of f (z), it is also a pole of
f (z) only when it is a multiple pole of f (z).

Let the multiplicity of the pole for f (z) be ?i; then for very large values
of z we have

/(*)-*•*£),

where <j> is holomorphic for very large values of z and does not vanish at
infinity ; hence

A«)**" •*-*'•

44.] UNIFORM FUNCTION 69

The coefficient of zn~* is holomorphic for very large values of z and does not
vanish at infinity ; hence infinity is a pole of/' (z} of multiplicity n — 1.

If n be unity, so that infinity is a simple pole of / (z), then it is not a
pole of/' (2); the derivative is then finite at infinity.

45. THEOREM VIII. A function, which has no singularity in a finite
part of the plane, and has z = oo for a pole, is an algebraical polynomial.

Let n, necessarily a finite integer, be the order of multiplicity of the pole
at infinity : then the function / (z) can be expressed in the form

1 + ...... +an^z + Q - ,

\zJ

where Q (- J is a holomorphic function for very large values of z, and is finite
(or zero) when z is infinite.

Now the first n terms of the series constitute a function which has no
singularities in the finite part of the plane : and / (z) has no singularities

in that part of the plane. Hence Q ( - J has no singularities in the finite part

of the plane : it is finite for infinite values of z. It thus can never have an
infinite value: and it is therefore merely a constant, say an. Then

/ (z) = a,zn + a^-1 + ...... + an^z + an,

a polynomial of degree equal to the multiplicity of the pole at infinity,
supposed to be the only pole of the function.

46. The above result may be obtained in the following manner.

Since z = GO is a pole of multiplicity n, the limit of z~nf (z} is not infinite
when z = oo .

Now in any finite part of the plane the function is everywhere finite, so
that we can use the expansion

where £ = *'""> dt

''+l t-z'

the integral being taken round a circle of any radius r enclosing the point z
and having its centre at the origin. As the subject of integration is finite
everywhere along the circumference, we have, by Darboux's expression in
(IV.) S 14,

T»i T _ z

where r is some point on the circumference and X is a quantity of modulus
not greater than unity.

70 TRANSCENDENTAL AND [46.

Let T = reia- ; then

X . fM

"• 71-4-1 °flii »/ \ /

'?* rn

r

f(T\
By definition, the limit of n as T (and therefore r) becomes infinitely

(£ -\—1
1 -- e~ai } is unity.
r J

Since \ is not greater than unity, the limit of \jr in the same case is zero ;
hence with indefinite increase of r, the limit of R is zero and so

shewing as before that/(^) is an algebraical polynomial.

47. As the quantity n is necessarily a positive integer*, there are two
distinct classes of functions discriminated by the magnitude of n.

The first (and the simpler) is that for which n has a finite value. The
polynomial then contains only a finite number of terms, each with a positive
integral index ; and the function is then a rational, integral, algebraical
polynomial of degree n.

The second (and the more extensive, as significant functions) is that
for which n has an infinite value. The point z = oo is not a pole, for then
the function does not satisfy the test of § 42 : it is an essential singularity
of the function, which is expansible in an infinite converging series
of positive integral powers. To functions of this class the general term
transcendental is applied.

The number of zeros of a function of the former class is known : it is
equal to the degree of the function. It has been proved that the zeros of a
transcendental function are isolated points, occurring necessarily in finite
number in any finite part of the region of continuity of the function, no
point on the boundary of the part being infinitesimally near an essential
singularity ; but no test has been assigned for the determination of the total
number of zeros of a function in an infinite part of the region of con
tinuity.

Again, when the zeros of a polynomial are given, a product-expression can
at once be obtained that will represent its analytical value. Also we know
that, if a be a zero of any uniform analytic function of multiplicity n, the
function can be represented in the vicinity of a by the expression

(x-a}n<t>(z\

where <£ (z) is holomorphic in the vicinity of a. The other zeros of the
function are zeros of <f> (z) ; this process of modification in the expression

* It is unnecessary to consider the zero value of n, for the function is then a polynomial of
order zero, that is, it is a constant.

47.] ALGEBRAICAL UNIFORM FUNCTIONS 71

can be continued for successive zeros so long as the number of zeros taken
account of is limited. But when the number of zeros is unlimited, then the
inferred product-expression for the original function is not necessarily a
converging product; and thus the question of the formal factorisation of a
transcendental function arises.

48. THEOREM IX. A function, all the singularities of which are accid
ental, is a rational, algebraical, meromorphic function.

Since all the singularities are accidental, each must be of finite
multiplicity ; and therefore infinity, if an accidental singularity, is of finite
multiplicity. All the other poles are in the finite part of the plane ; they
are isolated points and therefore only finite in number, so that the total
number of distinct poles is finite and each is of finite order. Let them be
«!, a2, ...... , a^ of orders m1} m2, ...... , m^ respectively : let m be the order of

the pole at infinity: and let the poles be arranged in the sequence of
decreasing moduli such that [aj > aF_! > ...... >|&i|-

Then, since infinity is a pole of order m, we have

/ 0) = amzm + a^z™-1 + ...... + a^z + /„ <»,

where /„ (z) is not infinite for infinite values of z. Now the polynomial

m

Sttj^ is not infinite for any finite value of z ; hence f0 (z) is infinite for all

i = l

the finite infinities of f (z) and in the same way, that is, the function f0(z)
has «!, ...... , a^ for its poles and it has no other singularities.

Again, since «M is a finite pole of multiplicity WM, we have

where fi(z) is not infinite for z = all and, as f0(z) is not infinite for z=<x> ,
evidently f^ (z) is not infinite for z = oo . Hence the singularities of f^ (z) are
merely the poles a1} ...... , aF_i ; and these are all its singularities.

Proceeding in this manner for the singularities in succession, we ultimately
reach a function f^ (z) which has only one pole a^ and no other singularity,
so that

k k

where g (z) is not infinite for z = a^ But the function f^(z) is infinite only
for 2 = 0,!, and therefore g (2) has no infinity. Hence g (z} is only a constant,
say k0 : thus

9 (*} = ^o-

Combining all these results we have a, finite number of series to add together:
and the result is that

72 UNIFORM [48.

where g1 (z) is the series k0 + a-^z + + amzm, and \ I is the sum of the

finite number of fractions. Evidently gs (z) is the product

{z — Oi)m> (z — a2)ma (z — aM)mfx ;

and g» (z) is at most of degree

If F (z} denote g1 (z} g3 (z) + g^ (z), the form of / (z) is

</.(*)'

that is, f (z) is a rational, algebraical, meromorphic function.

It is evident that, when the function is thus expressed as an algebraical
fraction, the degree of F (z) is the sum of the multiplicities of all the poles
when infinity is a pole.

COROLLARY I. A function, all the singularities of which are accidental,
has as many zeros as it has accidental singularities in the plane.

If z = oo be a pole, then it follows that, because f(z) can be expressed
in the form

it has as many zeros as F(z), unless one such should be also a zero of g^(z).
But the zeros of g3(z) are known, and no one of them is a zero of F(z), on
account of the form of f(z} when it is expressed in partial fractions. Hence
the number of zeros off(z) is equal to the degree of F(z}, that is, it is equal
to the number of poles off(z}.

If 2=00 be not a pole, two cases are possible; (i) the function f (z) may be
finite for z = oo , or (ii) it may be zero for z = oo . In the former case, the
number of zeros is, as before, equal to the degree of F (z), that is, it is equal
to the number of infinities.

In the latter case, if the degree of the numerator F (z) be K less than
that of the denominator gs (z), then z = oo is a zero of multiplicity K ; and it
follows that the number of zeros is equal to the degree of the numerator
together with K, so that their number is the same as the number of accidental
singularities.

COROLLARY II. At the beginning of the proof of the theorem of the
present section, it is proved that a function, all the singularities of which are
accidental, has only a finite number of such singularities.

Hence, by the preceding Corollary, such a function can have only a finite
number of zeros.

If, therefore, the number of zeros of a function be infinite, the function
must have at least one essential singularity.

48.] ALGEBRAICAL FUNCTIONS 73

COROLLARY III. When a uniform analytic function has no essential
singularity, if the (finite) number of its poles, say clv.., cm, be m, no one of
them being at z = oo , and if the number of its zeros, say aly..., am, be also m,
no one of them being at z = oo , then the function is

„ n

* a

r=l \Z - CT

except possibly as to a constant factor.

When z = oo is a zero of order n, so that the function has m — n zeros, say
«i, a2,..., in the finite part of the plane, the form of the function is

m-n

II (z — ar)

r=l

r=l

and, when z = <x> is a pole of order p, so that the function has m - p poles,
say cl} c.2>..., in the finite part of the plane, the form of the function is

II (Z - Or)

r=l _

m-p ~

COROLLARY IV. All the singularities of rational algebraical meromorphic
functions are accidental.

CHAPTER V.

TRANSCENDENTAL INTEGRAL FUNCTIONS.

49. WE now proceed to consider the properties of uniform functions
which have essential singularities.

The simplest instance of the occurrence of such a function has already
been referred to in § 42 ; the function has no singularity except at z = oo ,
and that value is an essential singularity solely through the failure of the
limitation to finiteness that would render the singularity accidental. The
function is then an integral function of transcendental character ; and it is
analytically represented (§ 26) by G (z) an infinite series in positive powers of
z, which converges everywhere in the finite part of the plane and acquires
an infinite value at infinity alone.

The preceding investigations shew that uniform functions, all the singu
larities of which are accidental, are rational algebraical functions — their
character being completely determined by their uniformity and the accidental
nature of their singularities, and that among such functions having the same
accidental singularities the discrimination is made, save as to a constant
factor, by means of their zeros.

Hence the zeros and the accidental singularities of a rational algebraical
function determine, save as to a constant factor, an expression of the function
which is valid for the whole plane. A question therefore arises how far
the zeros and the singularities of a transcendental function determine the
analytical expression of the function for the whole plane.

50. We shall consider first how far the discrimination of transcendental
integral functions, which have no infinite value except for z = oc , is effected
by means of their zeros*.

* The following investigations are based upon the famous memoir by Weierstrass, " Zur
Theorie der eindeutigen analytischen Functionen," published in 187G : it is included, pp. 1 — 52,
in the Abhandlungen aus der Functioiienlehre (Berlin, 1886).

In connection with the product-expression of a transcendental function, Cayley, " Memoire sur
les fonctions doublement periodiques," Liouville, t. x, (1845), pp. 385 — 420, or Collected Works,
vol. i, pp. 156 — 182, should be consulted.

50.]

CONVERGING INFINITE PRODUCTS

75

Let the zeros aly a2, a3,... be arranged in order of increasing moduli; a
finite number of terms in the series may have the same value so as to allow
for the existence of a multiple zero at any point. After the results stated
47, it will be assumed that the number of zeros is infinite ; that,

n

subject to limited repetition, they are isolated points ; and, in the present
chapter, that, as n increases indefinitely, the limit of \an\ is infinity. And it
will be assumed that at\ > 0, so that the origin is temporarily excluded from
the series of zeros.

Let z be any point in the finite part of the plane. Then only a limited
number of the zeros can lie within and on a circle centre the origin and
radius equal to \z\ ; let these be a]5 a2,..., afc_1} and let ar denote any one of
the other zeros. We proceed to form the infinite product of quantities ur,
where ur denotes

and gr is a rational integral function of z which, being subject to choice, will
be chosen so as to make the infinite product converge everywhere in the
plane. We have

00 \

w=l

a series which converges because \z < \ar\. Now let

ffr =

then

«> 1 / ^ \n
i v -1- / •* \

logi<r = - 2 -f£J ,

»j = S »4 \**rr

and therefore

Hence

•-— "

if the expression on the right-hand side be finite, that is, if the series

oo ce I / _ \ n

2 S -(-)

r=ftw=«^ \flrf

converge unconditionally. Denoting the modulus of this series by M, we
have

z
a,.

00 00 1

M < 2 2 -

r-k n=s M

SO that

sM< S 2

r=k n=s

7G

WEIERSTRASS'S CONVERGING

[50.

whence since 1 - — is the smallest of the denominators in terms of the last

«*
sum, we have

sM\l-

z

[ <

00

Z

8

1

«&

j r=k

ar

• I l

•

*-l

If, as is not infrequently the case, there be any finite integer s for which (and
therefore for all greater indices) the series

2 1

Is '

00

and therefore the series 2 \ar\-s, converges, we choose s to be that least

r=k

integer. The value of M then is finite for all finite values of z ; the series

oo co T / ~\n

2 2 - -

n

r=k

converges unconditionally and therefore
is a converging product when

Let the finite product

A-l (/ f

n |(i--

m=l l\ am

be associated as a factor with the foregoing infinite converging product. Then
the expression

oo ( f 2 \ 2

T-=I (\ ar/
is an infinite product, converging uniformly and unconditionally for all finite

00

values of z, provided the finite integer s be such as to make the series 2

converge uniformly and unconditionally.

Since the product converges uniformly and unconditionally, no product
constructed from its factors ur, say from all but one of them, can be infinite.
Now the factor

"5?i/£-Y

\ ?L\en=\n\am)

vanishes for z = am; hence f(z) vanishes for z = am. Thus the function,
evidently uniform after what has been proved, has the assigned points
Oj, a2)... and no others for its zeros.

50.]

INFINITE PRODUCT

77

Further, z = oo is an essential singularity of the function ; for it is an
essential singularity of each of the factors on account of the exponential
element in the factor.

51. But it may happen that no finite integer s can be found which will
make the series

00

r=l

converge*. We then proceed as follows.

Instead of having the same index s throughout the series, we associate
with every zero ar an integer mr chosen so as to make the series

n=l @"n \Q"n

a converging series. To obtain these integers, we take any series of decreasing
real positive quantities e, e1} e2,..., such that (i) e is less than unity and
(ii) they form an unconditionally converging series ; and we choose integers
ftir such that

These integers make the foregoing series of moduli converge. For,
neglecting the limited number of terms for which \z\^ a\, and taking e
such that

z

we have for all succeeding terms

and therefore

ar

Hence, except for the first k — 1 terms, the sum of which is finite, we have

n=k

which is finite because the series

... converges. Hence the series

n=l

s a converging series.

* For instance, there is no finite integer s that can make the infinite series
(log 2)-' + (log 3)- + (log 4)- + . . .

converge. This series is given in illustration by Hermite, Cours a la faculte des Sciences (4mc ed.
1891), p. 86.

78

WEIERSTRASS'S CONVERGING

[51.

Just as in the preceding case a special expression was formed to serve as
a typical factor in the infinite product, we now form a similar expression
for the same purpose. Evidently

1 - a; = ei<* a-*) = e
if \x\ < 1. Forming a function E (x, m) denned by the equation

m xr

S -

E (x, m)=(l-x)e r=1 r ,

we have E (x, m) =

In the preceding case it was possible to choose the integer m so that it
should be the same for all the factors of the infinite product, which was

0

ultimately proved to converge. Now, we take x = — and associate mn as
the corresponding value of m. Hence, if

/(*) =

where

< \z < |ttjfc|, we have

n=k

- s s

The infinite product represented by f(z) will converge if the double series in
the exponential be a converging series.

Denoting the double series by S, we have

\S\<* 2

2^*
2

n=kr=l

r+mn

< 2

n—k

1+TOM

1 4£

\an

on effecting the summation for r. Let A be the value of 1 —

all the remaining values of n we have

1

z !

- ; then for

— >>A,

and so

n=/fc

This series converges; hence for finite values of z\ the value of \S\ is
finite, so that S is a converging series. Hence it follows that f(z) is an

51.] INFINITE PRODUCT 79

unconditionally converging product. We now associate with f(z) as factors
the k — I functions

for i= 1, 2,..., k—1; their number being finite, their product is finite and
therefore the modified infinite product still converges. We thus have

an unconditionally converging product.

Since the product G (z) converges unconditionally, no product constructed
from its factors E, say from all but one of them, can be infinite. The factor

vanishes for the value z = an and only for this value ; hence G (z) vanishes for
z = an. It therefore appears that G(z) has the assigned points a1} a.,, a3, ...
and no others for its zeros ; and from the existence of the exponential in each
of the factors it follows that z = oo is an essential singularity of the factor and
therefore it is an essential singularity of the function.

Denoting the series in the exponential by gn (z\ so that

mn 1 / ~ \ r

*<*>-£?(£)•

71 / z \ i-. Z\

we have A — , mn = 1 e^' ;

\an / V aJ

and therefore the function obtained is

; G (z)= H \(l — — ] eg«(zl

n = l (\ Q"n,l

The series gn usually contains only a limited number of terms ; when the
number of terms increases without limit, it is only with indefinite increase
of | an | and the series is then a converging series.

It should be noted that the factors of the infinite product G (z) are the
expressions E no one of which, for the purposes of the product, is resoluble
into factors that can be distributed and recombined with similarly obtained
factors from other expressions E; there is no guarantee that the product
of the factors, if so resolved, would converge uniformly and unconditionally,
and it is to secure such convergence that the expressions E have been
constructed.

It was assumed, merely for temporary convenience, that the origin was not
a zero of the required function ; there obviously could not be a factor of
exactly the same form as the factors E if a were the origin.

80 TRANSCENDENTAL INTEGRAL FUNCTION [51.

If, however, the origin were a zero of order X, we should have merely
to associate a factor ZK with the function already constructed.

We thus obtain Weierstrass's theorem : —

It is possible to construct a transcendental integral function such that it
shall have infinity as its only essential singularity and have the origin (of
multiplicity X), a^, az, a3, ... as zeros ; and such a function is

00 ( / z\

ZK n ui — U^

n=i

where gn(z) is a rational, integral, algebraical function of z, the form of which
is dependent upon the law of succession of the zeros.

52. But, unlike uniform functions with only accidental singularities, the
function is not unique : there are an unlimited number of transcendental
integral functions with the same series of zeros and infinity as the sole essential
singularity, a theorem also due to Weierstrass.

For, if G! (z) and G (z) be two transcendental, integral functions with the
same series of zeros in the same multiplicity, and z = oo as their only essential
singularity, then

G(z}
is a function with no zeros and no infinities in the finite part of the plane.

Denoting it by £r2, then

1 ^

<72 dz

is a function which, in the finite part of the plane, has no infinities; and
therefore it can be expanded in the form

a series converging everywhere in the finite part of the plane. Choosing a
constant C0 so that 6r2 (0) = e*7", we have on integration

where g(z) = C0

and g (z) is finite everywhere in the finite part of the plane. Hence it follows
that, ifg(z) denote any integral function of z which is finite everywhere in the
finite part of the plane, and if G (z) be some transcendental integral function
with a given series of zeros and z= oo as its sole essential singularity, all
transcendental integral functions with that series of zeros and z= <x> as the
sole essential singularity are included in the form

£(*)«*».

COROLLARY I. A function which has no zeros in the finite part of the
plane, no accidental singularities and z=<x> for its sole essential singularity
is necessarily of the form

52.] AS AN INFINITE PRODUCT 81

where g (z) is an integral function of z finite everywhere in the finite part
of the plane.

COROLLARY II. Every transcendental function, which has the same zeros
in the same multiplicity as an algebraical polynomial A (z) — the number,
therefore, being necessarily finite — , ivhich has no accidental singularities and
has z = oo for its sole essential singularity, can be expressed in the form

A (z)

COROLLARY III. Every function, which has an assigned series of zeros
and an assigned series of poles and has z = oo for its sole essential singu
larity, is of the form

where the zeros of G0(z) are the assigned zeros and the zeros of Gp(z) are the
assigned poles.

For if Op (z) be any transcendental integral function, constructed as in
the proposition, which has as its zeros the poles of the required function in
the assigned multiplicity, the most general form of that function is

0p(*)e*»,

where h (z) is integral. Hence, if the most general form of function which
has those zeros for its poles be denoted by f(z), we have

f(z)Gp(z)e^

as a function with no poles, with infinity as its sole essential singularity, and
with the assigned series of zeros. But if G0 (z) be any transcendental integral
function with the assigned zeros as its zeros, the most general form of function
with those zeros is

and so f(z) Gp (z) eh ® = G0 (z) e° & ,

whence / (z) = ?$1 effW,

Lrp (z)

in which g (z) denotes g (z) — h (z).

If the number of zeros be finite, we evidently may take G0(z) as the
algebraical polynomial with those zeros as its only zeros.

If the number of poles be finite, we evidently may take Gp(z) as the
algebraical polynomial with those poles as its only zeros.

And, lastly, if a function have a finite number of zeros, a finite number
of accidental singularities and 2=00 as its sole essential singularity, it can
be expressed in the form

F.

82 PRIMARY [52.

where P and Q are rational integral polynomials. This is valid even though
the number of assigned zeros be not the same as the number of assigned
poles ; the sole effect of the inequality of these numbers is to complicate the
character of the essential singularity at infinity.

53. It follows from what has been proved that any uniform function,
having z = <x> for its sole essential singularity and any number of assigned
zeros, can be expressed as a product of expressions of the form

a

Such a quantity is called* a primary factor of the function.

It has also been proved that : —

(i) If there be no zero an, the primary factor has the form

(ii) The exponential index gn (z) may be zero for individual primary
factors, though the number of such factors must, at the utmost,
be finite f.

(iii) The factor takes the form z when the origin is a zero.
Hence we have the theorem, due to Weierstrass : —

Every uniform integral function of z can be expressed as a product of
primary factors, each of the form

(kz + I) e3W,

where g(z) is an appropriate integral function of z vanishing with z and where
k, I are constants. In particular factors, g (z) may vanish ; and either k or I,
but not both k and I, may vanish with or without a non-vanishing exponential
index g(z).

54. It thus appears that an essential distinction between transcendental
integral functions is constituted by the aggregate of their zeros : and we may
conveniently consider that all such functions are substantially the same when
they have the same zeros.

There are a few very simple sets of functions, thus discriminated by their
zeros: of each set only one member will be given, and the factor e^(z}, which
makes the variation among the members of the same set, will be neglected
for the present. Moreover, it will be assumed that the zeros are isolated
points.

I. There may be a finite number of zeros ; the simplest function is then
an algebraical polynomial.

* Weierstrass's term is Prim/unction, I.e., p. 15.

t Unless the class (§ 59) be zero, when the index is zero for all the factors.

54.] FACTORS 83

II. There may be a singly-infinite system of zeros. Various functions
will be obtained, according to the law of distribution of the zeros.

Thus let them be distributed according to a law of simple arithmetic
progression along a given line. If a be a zero, co a quantity such that co \
is the distance between two zeros and arg. co is the inclination of the line,
we have

a + mco,

for integer values of m from - oo to + oo , as the expression of the series of
the zeros. Without loss of generality we may take a at the origin — this
is merely a change of origin of coordinates — and the origin is then a
simple zero : the zeros are given by mco, for integer values of m from
— oo to + oo .

Now 2 — - = - 2 — is a diverging series ; but an integer s — the lowest

value is s = 2 — can be found for which the series S I - ] converges uni-

\mcoj
formly and unconditionally. Taking s = 2, we have

, . '-1 1 / z \n z
ffm (z) = 2 - — = — ,
»=i n vW m™
so that the primary factor of the present function is

Z \

--- )
mco/

m<a

e

and therefore, by § 52, the product

/«-,SJ(i- *-)

-oo (\ mcoj
converges uniformly and unconditionally for all finite values of z.

The term corresponding to m = 0 is to be omitted from the product ; and
it is unnecessary to assume that the numerical value of the positive infinity
for m is the same as that of the negative infinity for m. If, however, the
latter assumption be adopted, the expression can be changed into the ordinary
product-expression for a sine, by combining the primary factors due to values
of m that arc equal and opposite : in fact, then

co . TTZ
= - - sin — .

7T CO

This example is sufficient to shew the importance of the exponential term in the
primary factor. If the product be formed exactly as for an algebraical polynomial, then
the function is

z n

in the limit when both p and q are infinite. But this is known* to be

- ) - sin — .

77 0)

* Hobson's Trigonometry, § 287.

6—2

84 PRIMARY [54.

Another illustration is afforded by Gauss's II-function, which is the limit when k is
infinite of

1.2.3 ...... k

(«+!) (0+2) ...... (z+k)

This is transformed by Gauss* into the reciprocal of the expression

that is, of (1 +*) jj {(l +^) e "2l°g

the primary factors of which have the same characteristic form as in the preceding
investigation, though not the same literal form.

It is chiefly for convenience that the index of the exponential part of the primary

t-l 1/2 \n

factor is taken, in § 50, in the form 2 - ( — ) . With equal effectiveness it may be

n=l % \~^T /
»-l 1

taken in the form 2 - br nzn. provided the series
'

r=k «=i n
converge uniformly and unconditionally.

Ex. 1. Prove that each of the products

form=+l, ±3, +5, ...... to infinity, and

the term for n = Q being excluded from the latter product, converges uniformly and uncon
ditionally and that each of them is equal to cos z. (Hermite and Weyr.)

Ex. 2. Prove that, if the zeros of a transcendental integral function be given by the
series

0) +&>, ±4w, +9cB, ...... to infinity,

the simplest of the set of functions thereby determined can be expressed in the form

( fz\*\ , (. fz\*\
sm X?r I - }- sin -UTT - ) }- .
I W ) ( W J

Ex. 3. Construct the set of transcendental integral functions which have in common
the scries of zeros determined by the law m2a>l + 2m<a2 + a>3 for all integral values of m
between - oo and + oo ; and express the simplest of the set in terms of circular functions, j

55. The law of distribution of the zeros, next in importance and sub
stantially next in point of simplicity, is that in which the zeros form a doubly-
infinite double arithmetic progression, the points being the oo 2 intersections
of one infinite system of equidistant parallel straight lines with another
infinite system of equidistant parallel straight lines.

The origin may, without loss of generality, be taken as one of the zeros.
If a) be the coordinate of the nearest zero along the line of one system
passing through the origin, and &>' be the coordinate of the nearest zero along

* Ges. Wcrke, t. Hi, p. 145; the example is quoted in this connection by Weierstrass, I.e., !
p. 15.

55.] FACTIOUS 85

the line of the other system passing through the origin, then the complete
series of zeros is given by

fl = mw + mm,

for all integral values of m and all integral values of ni between — <x> and
+ oo . The system of points may be regarded as doubly -periodic, having &>
arid &>' for periods.

It must be assumed that the two systems of lines intersect. Other
wise, w and to' would have the same argument and their ratio would be a real
quantity, say a ; and then

ft

— = m + m a.

CO

Whether a be commensurable or incommensurable, the number of pairs
of integers, for which m + in' a. is zero or may be made less than any small
quantity 8, is infinite ; and in either case we should have the origin a zero
for each such pair, that is, altogether the origin would be a zero of infinite
multiplicity. This property of a function is to be considered as excluded,
for it would make the origin an essential singularity instead of, as required,
an ordinary point of the transcendental integral function. Hence the ratio of
the quantities w and w' is not real.

56. For the construction of the primary factor, it is necessary to render
the series

converging, by appropriate choice of integers sm>m. It is found to be
possible to choose an integer s to be the same for every term of the series,
corresponding to the simpler case of the general investigation, given in § 50.
As a matter of fact, the series

diverges for s = I (we have not made any assumption that the positive and
the negative infinities for m are numerically equal, nor similarly as to m') ;
the series converges for s = 2, but its value depends upon the relative values
of the infinities for m and m'; and s = 3 is the lowest integral value for which,
as for all greater values, the series converges uniformly and unconditionally.

There are various ways of proving the uniform and unconditional conver
gence of the series 2ft~M when /* > 2 : the following proof is based upon a
general method due to Eisenstein*.

»I=QO n=oo

First, the series S 2 (m2 + n*)~* converges uniformly and uricondi-

m= — «> n= -oo

tionally, if /j,> 1. Let the series be arranged in partial series : for this purpose,

Crelle, t. xxxv, (1847), p. 161 ; a geometrical exposition is given by Halphen, Traite des
fonctions elliptiques, t. i, pp. 358 — 362.

86 WEIERSTRASS'S FUNCTION AS [56.

we choose integers k and I, and include in each such partial series all
the terms which satisfy the inequalities

m ^ 2*+1,

so that the number of values of m is 2* and the number of values of n is 2*.
Then, if k + I = %K, we have

so that each term in the partial series ^ ^- . The number of terms in the

^" J*

partial series is 2fc . 2*, that is, 22K : so that the sum of the terms in the
partial series is

Take the upper limit of k and I to be p, ultimately to be made infinite.
Then the sum of all the partial series is

which, when p = oo , is a finite quantity if p > 1.
Next, let (a = a. + /3i, «' = 7 + Si, so that

ft = mw + nay' = ma + ny + i (m{3 + n8) ;
hence, if 6 =• ma. + nj, (j> = m(3 + n$,

we have | ft 2 = fr + </>2.

Now take integers r and s such that

r<0<r + \, s<(jxs + ~L.

The number of terms ft satisfying these conditions is definitely finite and is
independent of m and n. For since

m(«S —

n a -

and a8 — (3y does not vanish because o>'/a> is not purely real, the number of
values of in is the integral part of

(r + 1)8 — sy

a.8 — fiy
less the integral part of

r8 — (s + 1 ) 7
a.8 — fly

that is, it is the integral part of (7 + 8)/(«8 — #7). Similarly, the number of
values of n is the integral part of (a + /3)/(aS - j3j). Let the product of the

56.] A DOUBLY-INFINITE PRODUCT 87

last two integers be q ; then the number of terms fl satisfying the in
equalities is q.

Then 22 1 ft \~* = 22 (&> + p)~*

< q 22 (r2 + s2)-'*,
which, by the preceding result, is finite when yu,> 1. Hence

22 (mco + m'(»)'}~-»-

converges uniformly and unconditionally when //, > 1 ; and therefore the least
value of s, an integer for which

22 (mco + m'co')~s
converges uniformly and unconditionally, is 3.

The series 22(?tto) + m'<»')~2 has a finite sum, the value of which depends* upon
the infinite limits for the summation with regard to m and m'. This dependence is
inconvenient and it is therefore excluded in view of our present purpose.

Ex. Prove in the same manner that the series

the multiple summation extending over all integers mlt m2, ...... , mn between — oo and

+ oo , converges uniformly and unconditionally if 2/j.>n. (Eiseustein.)

57. Returning now to the construction of the transcendental integral
function the zeros of which are the various points H, we use the preceding
result in connection with § 50 to form the general primary factor. Since
s = 3, we have

s-l

and therefore the primary factor is

Moreover, the origin is a simple zero. Hence, denoting the required function
by a (z), we have

00 °°

<r(z) = zU H

— 00 -00

as a transcendental integral function which, since the product converges uni
formly and unconditionally for all finite values of z, exists and has a finite
value everywhere in the finite part of the plane; the quantity O denotes
mco + mV, and the double product is taken for all values of m and of m
between — oo and + oo , simultaneous zero values alone being excluded.

This function will be called Weierstrass's o-function ; it is of importance
in the theory of doubly-periodic functions which will be discussed in Chapter
XL

* See a paper by the author, Quart. Journ. of Math., vol. xxi, (1886), pp. 261—280.

88 PRIMARY FACTORS [57.

Ex. If the doubly-infinite series of zeros be the points given by

Q = m2^ + 2wm&>2 + «2o>3,

wi> W2) W3 being such complex constants that i2 does not vanish for real values of m and n,
then the series

2 2 Q-*

converges for s = 2. The primary factor is thus

and the simplest transcendental integral function having the assigned zeros is

The actual points that are the zeros are the intersections of two infinite systems of
parabolas.

58. One more result — of a negative character — will be adduced in this
connection. We have dealt with the case in which the system of zeros is a
singly-infinite arithmetical progression of points along one straight line and
with the case in which the system of zeros is a doubly-infinite arithmetical
progression of points along two different straight lines : it is easy to see that
a uniform transcendental integral function cannot exist with a triply -infinite
arithmetical progression of points for zeros.

A triply-infinite arithmetical progression of points would be represented
by all the possible values of

for all possible integer values for p1} p.,, p3 between — oo and + oc , where no
two of the arguments of the complex constants flj, H2, O3 are equal. Let

tlr = o)r + i(or', (r = 1, 2, 3) ;

then, as will be proved (§ 107) in connection with a later proposition, it is
possible* — and possible in an unlimited number of ways — to determine
integers plt p-2,ps so that, save as to infinitesimal quantities,

Pi _ _ £2 ___ PS

all the denominators in which equations differ from zero on account of the
fact that no two arguments of the three quantities fl1} H2, Ha are equal. For
each such set of determined integers we have

&.Qi+p&+p»to»

zero or infinitesimal, so that the origin is a zero of unlimited multiplicity or,
in other words, there is a space at the origin containing an unlimited number
of zeros. In either case the origin is an essential singularity, contrary to

* Jacobi, Oes. Werke, t. ii, p. 27.

58.] CLASS OF A FUNCTION 89

the hypothesis that the only essential singularity is for z — oo ; and hence a
uniform transcendental function cannot exist having a triply-infinite arith
metical succession of zeros.

59. In effecting the formation of a transcendental integral function by
means of its primary factors, it was seen that the expression of the primary
factor depends upon the values of the integers which make

a converging series. Moreover, the primary factors are not unique in form,
because any finite number of terms of the proper form can be added to the
exponential index in

and such terms will only the more effectively secure the convergence of the
infinite product. But there is a lower limit to the removal of terms with the
highest exponents from the index of the exponential ; for there are, in general,
minimum values for the integers m1} m»,..., below which these integers can
not be reduced, if the convergence of the product is to be secured.

The simplest case, in which the exponential must be retained in the
primary factor in order to secure the convergence of the infinite product, is
that discussed in § 50, viz., when the integers ml, w2)... are equal to one
another. Let m denote this common value for a given function, and let
m be the least integer effective for the purpose : the function is then said*
to be of class m, and the condition that it should be of class m is that the
integer m be the least integer to make the series

converge uniformly and unconditionally, the constants a being the zeros of
the function.

Thus algebraical polynomials are of class 0 ; the circular functions sin z
and cos z are of class 1 ; Wcierstrass's o--function, and the Jacobian elliptic
function sn z are of class 2, and so on : but in .no one of these classes do the
functions mentioned constitute the whole of the functions of that class.

60. One or two of the simpler properties of an aggregate of transcen
dental integral functions of the same class can easily be obtained.

Let a function f(z), of class n, have a zero of order r at the origin and

* The French word is genre ; the Italian is genere. Laguerre (see references on p. 92) appears
to have been the first to discuss the class of transcendental integral functions.

90

CLASS-PROPERTIES OF

[60.

have «!, a2)... for its other zeros, arranged in order of increasing moduli.
Then, by § 50, the function /O) can be expressed in the form

(*)='

M 1 / £\8

where </; (V) denotes the series 2 -f— 1 and G(z) must be properly deter
mined to secure the equality.

Now the series

is one which converges uniformly for all values of z that do not coincide with
one of the points a, that is, with one of the zeros of the original function.
For the sum of the series of the moduli of its terms is

1

Let d be the least of the quantities

1

, necessarily non-evanescent be

cause z does not coincide with any of the points a ; then the sum of the series

IS 1

which is a converging series since the function is of class n. Hence the
series of moduli converges and therefore the original series converges ; let it
be denoted by S (z), so that

1

=2

We have

Each step of this process is reversible in all cases in which the original pro-

f (z\
duct converges; if, therefore, it can be shewn of a function f(z) that -rr4

takes this form, the function is thereby proved to be of class n.
If there be no zero at the origin, the term - is absent.

CO.] TRANSCENDENTAL INTEGRAL JUNCTIONS 91

If the exponential factor G(z) be a constant so that G' (z) is zero, the
function /(.z) is said to be a simple function of class n.

61. There are one or two criteria to determine the class of a function :
the simplest of them is contained in the following proposition, due to
Laguerre*.

If, as z tends to the value <x> , a very great value of z can be found for

f'(z\
which the limit of z~n --jr\ , where f (z) is a transcendental, integral function,

J\z)
tends uniformly to the value zero, then f (z} is of class n.

Take a circle centre the origin and radius R, equal to this value of \z\\
then, by § 24, II., the integral

f'(t) dt

JL/lo!

SvtJ *»/(*)

taken round the circle, is zero when R becomes indefinitely great. But the
value of the integral is, by the Corollary in § 20,

' (t) 6A

+

!_ f<*> J./'_(0 Jfc_ _L y (
27ri J V- f(t) t-z 2-n-i <=1 J

tn f(fi t-Z 2-7TI J tn f(t) t-Z 2lri i=i J tn f(t} t-z'

taken round small circles enclosing the origin, the point z, and the points
a,i, which are the infinities of the subject of integration; the origin being
supposed a zero of /(t) of multiplicity r.

1 f» !/'(*) dt ._!/'(*)

JMOW

tnf(t}t-Z Znf(2}'
dt I I

»/ \^ /•

Shr»,

1 fWlf(t
iriJ «"/(0

L f<0> 1£(Q _^_ <^>(^) r
SwtJ tnf(t)t-z zn zn+*'

where ^> (^) denotes the integral, algebraical, polynomial

V " f +0 j~ i -f ~ if +•••'

when t is made zero. Hence

and therefore

which, by § GO, shews that/(V) is of class n.

* Comptcs Rendus, t. xciv, (1882), p. G36.

92 CLASS-PROPERTIES OF [61.

COROLLARY. The product of any finite number of functions of the same
class n is a function of class not higher than n ; and the class of the product
of any finite number of functions of different classes is not greater than the
highest class of the component functions.

The following are the chief references to memoirs discussing the class of functions :

Laguerrc, Comptes Rendus, t. xciv, (1882), pp. 160-163, pp. 635—638, ib. t. xcv, (1882),
pp. 828—831, ib. t. xcviii, (1884), pp. 79—81 ;

Poincare, Bull, des Sciences Math., t. xi, (1883), pp. 136—144 ;

Cesaro, Comptes Rendm, t. xcix, (1884), pp. 26—27, followed (p. 27) by a note by
Hermite; Giornale di Battaglini, t. xxii, (1884), pp. 191 — 200;

Vivanti, Giornale di Battaglini, t. xxii, (1884), pp. 243—261, pp. 378—380, ib. t. xxiii,
(1885), pp. 96—122, ib. t. xxvi, (1888), pp. 303—314 ;

Hermite, Cours d la faculte' des Sciences (4me ed., 1891), pp. 91 — 93.

Ex. 1. The function

2
1=1

where the quantities c are constants, n is a finite integer, and the functions J\ (z) are
algebraical polynomials, is of class unity.

Ex. 2. If a simple function be of class %, its derivative is also of class n.

Ex. 3. Discuss the conditions under which the sum of two functions, each of class n,
is also of class n.

Ex. 4. Examine the following test for the class of a function, due to Poincare.

Let a be any number, no matter how small provided its argument be such that eaz
vanishes when z tends towards infinity. Then / (z) is of class n, if the limit of

vanish with indefinite increase of z.

A possible value of a is 2 ciai~n~1, where C; is a constant of modulus unity.

Ex. 5. Verify the following test for the class of a function, due to de Sparre*.

Let X be any positive non-infinitesimal quantity ; then the function / (z) is of class n,
if the limit, for m = oo , of

\amn~l{\am + i\-\am\}
be not less than X. Thus sin z is of class unity.

Ex. 6. Let the roots of 0n + 1 = l be 1, a, a2, ...... , an; and let f (s) be a function

of class n. Then forming the product

n/(a«4

we evidently have an integral function of zn + 1; let it be denoted by F(zn + 1). The roots of
* Comptes Rendus, t. cii, (1886), p. 741.

61.] TRANSCENDENTAL INTEGRAL FUNCTIONS 93

F(zn+l) = Q are a^'for i=l, 2, and s = 0, 1, , n\ and therefore, replacing zn + 1 by z,

the roots ofF(z) = 0 are a?*1 for i=l, 2, .......

Since/ (z) is of class n, the series

converges uniformly and unconditionally. This series is the sum of the first powers of the
reciprocals of the roots of F(z}~ 0; hence, according to the definition (p. 89), F(z) is of
class zero.

It therefore follows that from, a function of any class a function of class zero with a
modified variable can be deduced. Conversely, by appropriately modifying the variable of
a given function of class zero, it is possible to deduce functions of any required class.

Ex. 7. If all the zeros of the function

=1 r anr
\
be real, then all the zeros of its derivative are also real. (Witting.)

00 I / ~ \

U\(l--)e'
«=* ^\ «W

CHAPTER VI.

FUNCTIONS WITH A LIMITED NUMBER OF ESSENTIAL SINGULARITIES.

62. SOME indications regarding the character of a function at an
essential singularity have already been given. Thus, though the function

is regular in the vicinity of such a point a, it may, like sn - at the origin,

%

have a zero of unlimited multiplicity or an infinity of unlimited multiplicity
at the point ; and in either case the point is such that there is no factor of
the form (z — a)x which can be associated with the function so as to make the
point an ordinary point for the modified function. Moreover, even when
the path of approach to the essential singularity is specified, the value
acquired is not definite : thus, as z approaches the origin along the axis of x,

so that its value may be taken to be 1 -f- (4>mK + x), the value of sn - is not

z

definite in the limit when m is made infinite. One characteristic of the
point is the indefiniteness of value of the function there, though in the
vicinity the function is uniform.

A brief statement and a proof of this characteristic were given in § 33 ;
the theorem there proved — that a uniform analytical function can assume
any value at an essential singularity — may also be proved as follows. The
essential singularity will be taken at infinity — a supposition that will be
found not to detract from generality.

Let f(z) be a function having any number of zeros and any number
of accidental singularities and £ = oo for its sole essential singularity ; then
it can be expressed in the form

/w-88*"'

where G1 (z) is algebraical or transcendental according as the number of zeros
is finite or infinite and G2(z) is algebraical or transcendental according as
the number of accidental singularities is finite or infinite.

If Cr2 (z) be transcendental, we can omit the generalising factor e°(z).
Then f(z) has an infinite number of accidental singularities ; each of them
in the finite part of the plane is of only finite multiplicity and therefore some
of them must be at infinity. At each such point, the function G2 (z) vanishes
and Ol (z) does not vanish ; and so f(z) has infinite values for z = oo .

62.] VALUE AT AN ESSENTIAL SINGULARITY 95

If Gz (2) be algebraical and Gl (z) be also algebraical, then the factor ea(z)
may not be omitted, for its omission would make f(z) an algebraical function.
Now z = oo is either an ordinary point or an accidental singularity of

ft <*)/<?.<*);

hence as g (z} is integral there are infinite values of z which make

infinite.

If G.>.(z) be algebraical and G^ (z) be transcendental, the factor eg(z) maybe
omitted. Let al5 a2,..., an be the roots of G2(z): then taking

f(z)= ^-

we have Ar=

a non-vanishing constant ; and so

where Gn (z) is a transcendental integral function. When 2 = oo , the value
of G3(z)/G.,(z) is zero, but Gn(z) is infinite ; hence f(z) has infinite values for

Z= 00 .

Similarly it may be shewn, as follows, that/(z) has zero values for 0 = oo .

In the first of the preceding cases, if Gl (z) be transcendental, so that f (z)
has an infinite number of zeros, then some of them must be at an infinite
distance; f(z) has a zero value for each such point. And if GI(Z) be
algebraical, then there are infinite values of z which, not being zeros of
G2(z), make f(z) vanish.

In the second case, when z is made infinite with such an argument as to
make the highest term in g(z) a real negative quantity, then f(z) vanishes
for that infinite value of z.

In the third case,/(V) vanishes for a zero of G1(z) that is at infinity.

Hence the value of f (z) for z= oo is not definite. If, moreover, there
be any value neither zero nor infinity, say G, which f(z) cannot acquire
for z = oo , then

/(*)-C

is a function which cannot be zero at infinity and therefore all its zeros are
in the finite part of the plane : no one of them is an essential singularity, for
f(z) has only a single value at any point in the finite part of the plane; hence
they are finite in number and are isolated points. Let H1 (z) be the alge
braical polynomial having them for its zeros. The accidental singularities
of f(z} — C are the accidental singularities of f(z) ; hence

96 FORM OF A FUNCTION NEAR [62.

where, if G2(z) be algebraical, the exponential h(z) must occur, since f(z),
and therefore f(z) — C, is transcendental. The function

-| sy / \

TJ1 ( ~\ — _ 2 \ / 0—h (z)

•* \*/ f t ~\ n TT f n\

J(z)-L H1 (z)

evidently has z= oo for an essential singularity, so that, by the second or
the third case above, it certainly has an infinite value for z = co , that is,
f(z) certainly acquires the value G for z= GO .

Hence the function can acquire any value at an essential singularity.

63. We now proceed to obtain the character of the expression of a
function at a point z which, lying in the region of continuity, is in the
vicinity of an essential singularity b in the finite part of the plane.

With b as centre describe two circles, so that their circumferences and
the whole area between them lie entirely within the region of continuity.
The radius of the inner circle is to be as small as possible consistent with
this condition; and therefore, as it will be assumed that b is the only
singularity in its own immediate vicinity, this radius may be made very
small.

The ordinary point z of the function may be taken as lying within the
circular ring-formed part of the region of continuity. At all such points in
this band, the function is holomorphic ; and therefore, by Laurent's Theorem
(§ 28), it can be expanded in a converging series of positive and negative
integral powers of z — b in the form

+ V-L(Z — 6)"1 + v2 (z — 6)~2 + . . .,
where the coefficients un are determined by the equation

un =

the integrals being taken positively round the outer circle, and the coefficients
vn are determined by the equation

the integrals being taken positively round the inner circle.

The series of positive powers converges everywhere within the outer circle
of centre b, and so (§ 26) it may be denoted by P (z - b) ; and the function P
may be either algebraical or transcendental.

The series of negative powers converges everywhere without the inner
circle of centre b ; and, since 6 is not an accidental but an essential singularity
of the function, the series of negative powers contains an infinite number of

63.] AN ESSENTIAL SINGULARITY 97

terms. It may be denoted by G I -- rh a series converging for all points

\z — o/

in the plane except z = b and vanishing when z — b = co.

Thus

is the analytical representation of the function in the vicinity of its essential
singularity b ; the function G is transcendental and converges everywhere in
tlie plane except at z =• b, and the function P, if transcendental, converges
uniformly and unconditionally for sufficiently small values of | z — b \ .

Had the singularity at b been accidental, the function G would have been
algebraical.

COROLLARY I. If the function have any essential singularity other than
b, it is an essential singularity of P (z — b) continued outside the outer circle ;

but it is not an essential singularity of G ( -- j] , for the latter function

\z — ol

converges everywhere in the plane outside the inner circle.

COROLLARY II. Suppose the function has no singularity in the plane
except at the point b ; then the outer circle can have its radius made infinite.
In that case, all positive powers except the constant term w0 disappear:
and even this term survives only in case the function have a finite value at
infinity. The expression for the function is

and the transcendental series converges everywhere outside the infinitesimal
circle round b, that is, everywhere in the plane except at the point b. Hence
the function can be represented by

This special result is deduced by Weierstrass from the earlier investiga
tions*, as follows. If f(z) be such a function with an essential singularity at
b, and if we change the independent variable by the relation

Z/==^b'

thcn/(V) changes into a function of z', the only essential singularity of which
is at / = GO . It has no other singularity in the plane ; and the form of the
function is therefore G (z'), that is, a function having an essential singularity
at b but no other singularity in the plane is

* Weierstrass (I.e.), p. 27.
F.

98 FORM OF A FUNCTION NEAR [63.

COROLLARY III. The most general expression of a function having its
sole essential singularity at b a point in the finite part of the plane and any
number of accidental singularities is

G,

where the zeros of the function are the zeros of GI, the accidental singularities
of the function are the zeros of G2, and the function g in the exponential is a
function which is finite everyiuhere except at b.

This can be derived in the same way as before ; or it can be deduced
from the corresponding theorem relating to transcendental integral functions,
as above. It would be necessary to construct an integral function G2(z')
having as its zeros

and then to replace z by - — j ; and G., is algebraical or transcendental,

Z 0

according as the number of zeros is finite or infinite.
Similarly we obtain the following result :

COROLLARY IV. A uniform function of z, which has its sole essential
singularity at b a point in the finite part of the plane and no accidental
singularities, can be represented in the form of an infinite product of primary
factors of the form

\z — b

which converges uniformly and unconditionally everywhere in the plane except
at z = b.

The function g ( =] is an integral function of T vanishing when

J \z-bj z-b

r vanishes; and k and I are constants. In particular factors, q( T)

z - b ^ \z - b)

may vanish ; and either k or I (but not both k and I) may vanish with or

without a vanishing exponent q { T ) .

J \z-bj

If tt{ be any zero, the corresponding primary factor may evidently be
expressed in the form

(z —

,z —

Similarly, for a uniform function of z with its sole essential singularity at b and
any number of accidental singularities, the product-form is at once derivable

63.] AN ESSENTIAL SINGULARITY 99

by applying the result of the present Corollary to the result given in
Corollary III.

These results, combined with the results of Chapter V., give the complete
general theory of uniform functions with only one essential singularity.

64. We now proceed to the consideration of functions, which have a
limited number of assigned essential singularities.

The theorem of § 63 gives an expression for the function at any point in
the band between the two circles there drawn.

Let c be such a point, which is thus an ordinary point for the function ;
then in the domain of c, the function is expansible in a form Pl (z — c).
This domain may extend to an essential singularity b, or it may be limited
by a pole d which is nearer to c than b is, or it may be limited by an
essential singularity / which is nearer to c than b is. In the first case, we
form a continuation of the function in a direction away from b; in the
second case, we continue the function by associating with the function
a factor (z — d)n which takes account of the accidental singularity ; in
the third case, we form a continuation of the function towards f. Taking
the continuations for successive domains of points in the vicinity of/, we can
obtain the value of the function for points on two circles that have / for
their common centre. Using these values, as in § 63, to obtain coefficients,
we ultimately construct a series of positive and negative powers converging
except at / for the vicinity of/ Different expressions in different parts
of the plane will thus be obtained, each being valid only in a particular
portion: the aggregate of all of them is the analytical expression of the
function for the whole of the region of the plane where the function exists.

We thus have one mode of representation of the function ; its chief
advantage is that it indicates the form in the vicinity of any point, though it
gives no suggestion of the possible modification of character elsewhere. This
deficiency renders the representation insufficiently precise and complete ; and
it is therefore necessary to have another mode of representation.

65. Suppose that the function has n essential singularities a,!, a*,..., an
and that it has no other singularity. Let a circle, or any simple closed
curve, be drawn enclosing them all, every point of the boundary as well
as the included area (with the exception of the n singularities) lying in
the region of continuity of the function.

Let z be any ordinary point in the interior of the circle or curve ; and
consider the integral , f/+\

I '*=-***>

taken round the curve. If we surround z and each of the n singularities by
small circles with the respective points for centres, then the integral round

7—2

100 FUNCTIONS WITH A LIMITED NUMBER [65.

the outer curve is equal to the sum of the values of the integral taken round
the n + l circles. Thus

and therefore

The left-hand side of the equation isf(z).
Evaluating the integrals, we have

where Gr is, as before, a transcendental function of - - — vanishing when

1

is zero.

z — ar

Now, of these functions, Gr{- -] converges everywhere in the plane

\& \jbfj

except at ar : and therefore, as n is finite,

r=i \z - a
is a function which converges everywhere in the plane except at the n points

Clj , . . . , an .

Because z = oc is not an essential singularity of f(z), the radius of the

circle in the integral =—. . ! /-- dt may be indefinitely increased. The value

ZTTI J s t — z

of f(t) tends, with unlimited increase of t, to some determinate value G which
is not infinite ; hence, as in § 24, II., Corollary, the value of the integral is
C. We therefore have the result that/0) can be expressed in the form

\z-a,
or, absorbing the constant C into the functions G and replacing the limitation,

that the function Gr(— — } shall vanish for — = 0, by the limitation

\z — arj z — ar

that, for the same value =0, it shall be finite, we have the theorem*:—

z — ar

If a given function f(z) have n singularities a^,..., an, all of which are in
the finite part of the plane and are essential singularities, it can be expressed
in the form

2G f-M,

r=i r \z - aj '

* The method of proof, by an integration, is used for brevity : the theorem can be established
by purely algebraical reasoning.

65.] OF ESSENTIAL SINGULARITIES 101

where Gr is a transcendental function converging everywhere in the plane

except at ar and having a determinate finite value gr for - — = 0, such

z — cir

n

that 2 gr is the finite value of the given function at infinity.
r=l

COROLLARY. If the given function have a singularity at oo , and n singu
larities in the finite part of the plane, then the function can be expressed in
the form

w / 1 \
G(z) + SG,(— L-J,

r=i \z-arr

where Gr is a transcendental or an algebraic polynomial function, according
as ar is an essential or an accidental singularity : and so also for G (z), accord
ing to the character of the singularity at infinity.

66. Any uniform function, which has an essential singularity at z = a,
can (§ 63) be expressed in the form

for points z in the vicinity of a. Suppose that, for points in this vicinity,
the function f(z) has no zero ; that it has no accidental singularity ; and
therefore, among such points z, the function

1 df(z)
/(*) dz

has no pole, and therefore no singularity except that at a which is essential.
Hence it can be expanded in the form

G(^+P(z-a\

z-a

where G converges everywhere in the plane except at a, and vanishes for
= 0. Let

z — a dz

, / 1 \

where 0^ I ^— ^ I converges everywhere in the plane except at a, and vanishes

for — — = o.

z — a

Then c, evidently not an infinite quantity, is an integer. To prove this,
describe a small circle of radius p round a : then taking z-a = pe91 so that

— = idd, we have
z — a

l M(*\

dz = P (z — a) dz

102 FUNCTIONS WITH A LIMITED NUMBER [66.

and therefore

Now JP(z — a)dz is a uniform function : and so is f(z). But a change
of 6 into 6 + 2-7T does not alter z or any of the functions : thus

actotr — 1 •

~~ *• i

and therefore c is an integer.

67, If the function /(z) have essential singularities alt..., an and no
others, then' it can be expressed in the form

n /I

C+ $9J-±

r=i \z-ar

If there be no zeros for this function f(z) anywhere (except of course such
as may enter through the indeterminateness at the essential singularities),
then

/(*) dz

has n essential singularities a1}..., an and no other singularities of any kind.
Hence it can be expressed in the form

n / 1 \
C+ 2 Gr(- -),
r=i \z-a,rl

where the function Gr vanishes with . Let

z — ar

cr d

I T~~

\js — a,./ z — ar dz { r\z — ar

where Gr I ) is a function of the same kind as Gr ( ) .

\z — ar/ \z — arj

Then all the coefficients cr, evidently not infinite quantities, are integers.
For, let a small circle of radius p be drawn round ar : then, if z — ar = peei, we
have

crdz

z — ar

= cri6,

and — ^ — = dPs (z - ar).

z — as

We proceed as before : the expression for the function in the former
case is changed so that now the sum 2Pg(0— ar) for 5 = !,..., i — 1,
r + 1,..., n is a uniform function; there is no other change. In exactly the
same way as before, we shew that every one of the coefficients cr is an
integer.

Hence it appears that if a given function f(z) have, in the finite part of

67.] OF ESSENTIAL SINGULARITIES 103

the plane, n essential singularities al,..., an and no other singularities and if
it have no zeros anywhere in the plane, then

f(z) dz

where all the coefficients c* are integers, and the functions G converge every
where in the plane except at the essential singularities and Gi vanishes for

-J-- 0.

Now, since f(z) has no singularity at oo , we have for very large values of z

and /'W = _>_

Z*

and therefore, for very large values of z,

_

f(z) dz u0 z2 z3

Thus there is no constant term in =7-^ ^r-^ , and there is no term in -. But

/(*) dz z

the above expression for it gives G as the constant term, which must therefore

vanish ; and it gives 2c; as the coefficient of - , for -7- •< (r< ( - — H will begin

z dz [ \z — ftj/ J

with — at least ; thus ^a must therefore also vanish.

Z"

Hence for a function f (z) which has no singularity at z= oo and no
zeros anywhere in the plane and of which the only singularities are the n
essential singularities at a1} a2,..., an, we have

/ (z) dz i=i z - Oi i=i dz ( \z- a
where the coefficients a are integers subject to the condition

n

2 ct = 0.

i=l

If an= oo , so that 2= GO is an essential singularity in addition to a2, a2,...,
an_j, there is a term 6r (z) instead of Gn( — - ] ; there is no term, that corre-

\Z — C^n/
/^

spends to - — , but there may be a constant G. Writing

—

z —

with the condition that G (z) vanishes when z — 0, we then have

-
= __ g ^

i=iz-at dz( v /J ».

104 PRODUCT-EXPRESSION OF [67.

where the coefficients d are integers, but are no longer subject to the
condition that their sum vanishes.

Let R* (z} denote the function

the product extending over the factors associated with the essential sin
gularities of f(z) that lie in the finite part of the plane; thus R*(z) is a
rational algebraical rneromorphic function. Since

1 dR*(z) = 2 d
R* (z) dz ~ i=\z — a,i'
we have

1 df(z) _ 1 dR*(z) =$ d_(-Q ( *
f(z) dz R* (z) dz i=\dz\ l\z — a^

where Gn ( — - ) is to be replaced by G (z) if an = <x> , that is, if z — oo be an
\z — anj

essential singularity off(z). Hence, except as to an undetermined constant
factor, we have

t=i

which is therefore an analytical representation of a function with n essential
singularities, no accidental singularities, and no zeros: and the rational alge
braical function R* (z) becomes zero or oo only at the singularities off(z).

If z = oo be not an essential singularity, then R* (z) for z = oo is equal to

M

unity because 2 Cf = 0.
1=1

COROLLARY. It is easy to see, from § 43, that, if the point a; be only an
accidental singularity, then a is a negative integer and wj I — - ) is zero : so

\Z — Oii/

that the polar property at c^ is determined by the occurrence of a factor
(z — a{)Ci solely in the denominator of the rational meromorphic function R* (z).

And, in general, each of the integral coefficients a is determined from the
expansion of the function f'(z) +f(z) in the vicinity of the singularity
with which it is associated.

68. Another form of expression for the function can be obtained from
the preceding; and it is valid even when the function has zeros not
absorbed into the essential singularities f.

Consider a function with one essential singularity, and let a be the
point ; and suppose that, within a finite circle of centre a (or within a finite
simple curve which encloses a), there are m simple zeros a, /3,..., X of the

+ See Guichard, TMorie des points singuliers essentiels, (These, Gauthier-Villars, Paris, 1883),
especially the first part.

68.] A FUNCTION 105

function f(z) — m being assumed to be finite, and it being also assumed that
there are no accidental singularities within the circle. Then, if

/(*) = (* - «) (z - /3). . .(* -\)F (z\

the function F (z) has a for an essential singularity and has no zeros within
the circle. Hence, for points z within the circle,

where (?, ( ----- ) converges everywhere in the plane and vanishes with - — ,

\z — a] z a

and P(z — a) is an integral function converging uniformly and unconditionally
within the circle ; moreover, c is an integer. Thus

F(z) = A(z- a)" eGl
Let (*-a)(*-/3)...(*-X) = (*-a

_ ( y _ r, \m

a)

then f(z) = (* - dTffl -- F(z}

\z u/

= A(z- ar+°gi (~}eG> ^ e ^~a] "z .
\z — ft/

Now of this product- expression for/(V) it should be noted: —

(i) That m + c is an integer, finite because m and c are finite :

<?,—-
(ii) The function e ' ^z~a' can be expressed in the form of a series con

verging uniformly and unconditionally everywhere, except at z = a, and

proceeding in powers of - — in the form

z a

....

z — a (z — af

It has no zero within the circle considered, for F (z) has no zero. Also gl(- - 1

\z a/

algebraical function of — ' — , beginning with unity and containing only

is an

z — a

a finite number of terms : hence, multiplying the two series together, we have

as the product a series proceeding in powers of - in the form

£ ~" a

— a

which converges uniformly and unconditionally everywhere outside any small
circle round a, that is, everywhere except at a. Let this series be denoted by

106 PRODUCT-EXPRESSION OF [68.

H I ]; it has an essential singularity at a and its only zeros are the

\z-aj

points a, (3,..., X, for the series multiplied by gl (— -) has

\z — ft/

no zeros :

(iii) The function fP (z — a) dz is a series of positive powers of z — a,
converging uniformly in the vicinity of a; and therefore Q^(z-d)dz can ke
expanded in a series of positive integral powers of z — a which converges
in the vicinity of a. Let it be denoted by Q (z — a) which, since it is a
factor of F (z), has no zeros within the circle.

Hence we have

/(*) = A (z - aYQ (z - a) H
where p, is an integer ; H ( — - J is a series that converges everywhere except

at a, is equal to unity when - vanishes, and has as its zeros the (finite)

z — a

number of zeros assigned to f(z) within a finite circle of centre a ; and
Q (z — a) is a series of positive powers of z — a beginning with unity which
converges — but has no zero — within the circle.

The foregoing function f(z) is supposed to have no essential singularity
except at ft. If, however, a given function have singularities at points
other than a, then the circle would be taken of radius less than the distance
of a from the nearest essential singularity.

Introducing a new function f{ (z} defined by the equation

the value of /[ (z) is Q (z — a) within the circle, but it is not determined by
the foregoing analysis for points without the circle. Moreover, as (z — a)*

and also Hi— ] are finite everywhere except possibly at a, it follows

that essential singularities of f(z) other than a must be essential singu
larities of fj (z). Also since /i (z) is Q(z — a) in the immediate vicinity of a,
this point is not an essential singularity of /i (z).

Thus /i (z) is a function of the same kind as f(z) ; it has all the essential
singularities of f(z) except ft, but it has fewer zeros, on account of the m

zeros of f(z) possessed by H ( - — ] . The foregoing expression for f(z) is

\Z — ft/

the one referred to at the beginning of the section.

If we choose to absorb into /x (z} the factors e \z~a' and e?P(z~° dz,
which occur in

(z - $*• ffl f Jil ^ (T- a)

\2 — ft/

68.] A FUNCTION 107

an expression that is valid within the circle considered, then we obtain a
result that is otherwise obvious, by taking

where now g± (— — ) is algebraical and has for its zeros all the zeros within

\Z — d/

the circle ; yu, is an integer; and/j (z) is a function of the same kind as f(z),
which now possesses all the essential singularities of f(z} but has zeros fewer

by the in zeros that are possessed by

z— a

69. Next, consider a function f(z) with n essential singularities al}
a2,..., an but without accidental singularities; and let it have any number of
zeros.

When the zeros are limited in number, they may be taken to be isolated
points, distinct in position from the essential singularities.

When the zeros are unlimited in number, then at least one of the
singularities must be such that an infinite number of the zeros lie within
a circle of finite radius, described round it as centre and containing no other
singularity. For if there be not an infinite number in such a vicinity
of some one point (which can only be an essential singularity, otherwise the
function would be zero everywhere), then the points are isolated and there
must be an infinite number at z = oo . If z = oo be an essential singularity, the
above alternative is satisfied : if not, the function, being continuous save at
singularities, must be zero at all other parts of the plane. Hence it follows
that if a uniform function have a finite number of essential singularities and
an infinite number of zeros, all but a finite number of the zeros lie within
circles of finite radii described round the essential singularities as centres ;
at least one of the circles contains an infinite number of the zeros, and some
of the circles may contain only a finite number of them.

We divide the whole plane into regions, each containing one but only one
singularity and containing also the circle round the singularity ; let the
region containing a{ be denoted by Ci, and let the region Gn be the part of
the plane other than Glt (72, ...... , Gn_^.

If the region G1 contain only a limited number of the zeros, then, by § 68,
we can choose a new function /i (z) such that, if

the function /j (z) has av for an ordinary point, has no zeros within the region
Glt and has a2, a3, ...... , an for its essential singularities.

If the region Cl contain an unlimited number of the zeros, then, as in
Corollaries II. and III. of § 63, we construct any transcendental function

108 GENERAL FORM OF A FUNCTION [69.

5xf— — ) , having a^ for its sole essential singularity and the zeros in GI for

\z — OiJ

all its zeros. When we introduce a function g: (2), defined by the equation

the function g:{z) has no zeros in GI and certainly has a2, a3, ...... , an for

essential singularities ; in the absence of the generalising factor of Glt it can
have Hi for an essential singularity. By § G7, the function ~g{ (2), defined by

gi (z) = 0 - cOc' ehl ^W ,

has no zero and no accidental singularity, and it has a^ as its sole essential
singularity : hence, properly choosing cx and hi, we may take

ft(*)-?i(*)/i(*)«

so that fi (z) does not have aj as an essential singularity, but it has all the
remaining singularities of ^ (z), and it has no zeros within C^.
In either case, we have a new function ft (z) given by

where /^ is an integer, the zeros off(z) that lie in GI are the zeros of GI ; the
function fi(z) has «2, »s> ...... > an (but not a^ for its essential singularities,

and it has the zeros of f(z) in the remaining regions for its zeros.
Similarly, considering (72, we obtain a function /2 (z), such that

where /A.2 is an integer, G2 is a transcendental function finite everywhere except
at a2 and has for its zeros all the zeros of ft (z) — and therefore all the zeros of
f (z) — that lie in G2 ; then f.2 (z) possesses all the zeros of f(z) in the regions
other than GI and C2, and has a3, a4,..., an for its essential singularities.

Proceeding in this manner, we ultimately obtain a function fn (z) which
has none of the zeros off(z) in any of the n regions GI, C2,..., Cn, that is, has
no zeros in the plane, and it has no essential singularities ; it has no acci

dental singularities, and therefore fn(z) is a constant. Hence, when we

•
substitute, and denote by S* (z} the product II (z — a^1, we have

Z —

as the most general form of a function having n essential singularities, no
accidental singularities, and any number of zeros. The function S* (z) is a
rational algebraical function of z, usually meromorphic inform, and it has the
essential singularities off(z) as its zeros and poles ; and the zeros of f (z) are
distributed among the functions Gt.

As however the distribution of the zeros by the regions C and therefore

69.] WITH ESSENTIAL SINGULARITIES 109

the functions G[ ) are somewhat arbitrary, the above form though general

\z — a]

is not unique.

If any one of the singularities, say am, had been accidental and not

essential, then in the corresponding form the function Gm ( - - ) would be

\Z — dm/

algebraical arid not transcendental.

70. A function f(z], which has any finite number of accidental singu
larities in addition to n assigned essential singularities and any number of
assigned zeros, can be constructed as follows.

Let A (z) be the algebraical polynomial which has, for its zeros, the
accidental singularities of f(z), each in its proper multiplicity. Then the
product

/(*)-A(*)

is a function which has no accidental singularities ; its zeros and its essential
singularities are the assigned zeros and the assigned essential singularities of
f (z) and therefore it is included in the form

n (
S*(z)U \0t

i=i (

where S* (z) is a rational algebraical meromorphic function having the points
Oi, a.,,..., an for zeros and poles. The form of the function f(z} is therefore

A

)}•

-ail)

71. A function f (z), which has an unlimited number of accidental singu
larities in addition to n assigned essential singularities and any number of
assigned zeros, can be constructed as follows.

Let the accidental singularities be of, /3',.... Construct a function f^ (z),
having the n essential singularities assigned to f (z}, no accidental singu
larities, and the series a!, /3',. . . of zeros. It will, by § 69, be of the form of a
product of n transcendental functions Gn+1,..., G.2n which are such that a
function G has for its zeros the zeros oif-i(z} lying within a region of the plane,
divided as in § 69 ; and the function Gn+t is associated with the point at-.

Thus / (z) = T*-(z) ft Gn

f=i

where T* (z) is a rational algebraical meromorphic function having its zeros
and its poles, each of finite multiplicity, at the essential singularities ofy(^).
Because the accidental singularities of f(z) are the same points and have
the same multiplicity as the zeros of /i (z), the function / (z) /x (z) has no
accidental singularities. This new function has all the zeros of f(z), and
al,...,an are its essential singularities; moreover, it has no accidental singu
larities. Hence the product f(z)fi (z) can be represented in the form

110 GENERAL FORM OF A FUNCTION [71.

and therefore we have

z-fi

(f-a)

as an expression of the function.

But, as by their distribution through the n selected regions of the plane
in § 69, the zeros can to some extent be arbitrarily associated with the
functions Gl} G*,,..., Gn and likewise the accidental singularities can to some
extent be arbitrarily associated with the functions Gn+l, Gn+»,..., G^i, the
product-expression just obtained, though definite in character, is not unique
in the detailed form of the functions which occur.

S* (z)
The fraction 7**) \

is algebraical and rational ; and it vanishes or becomes infinite only at the
essential singularities alt a.2,..., an, being the product of factors of the form
(z — «i)ms for i = l, 2,..., n. Let the power (z — a^ be absorbed into the
function G{/Gn+i for each of the n values of i ; no substantial change in the
transcendental character of Gi and of Gn+i is thereby caused, and we may
therefore use the same symbol to denote the modified function after the
absorption. Hence "f" the most general product-expression of a uniform
function of z which has n essential singularities al} a*,..., an, any unlimited
number of assigned zeros and any unlimited number of assigned accidental
singularities is

n ^

n —

\z-an

The resolution of a transcendental function with one essential singularity
into its primary factors, each of which gives only a single zero of the function,
has been obtained in § 63, Corollary IV.

We therefore resolve each of the functions G^..., Gm into its primary
factors. Each factor of the first n functions will contain one and only one zero
of the original functions / (.z) ; and each factor of the second n functions will
contain one and only one of the poles of f(z). The sole essential singularity
of each primary factor is one of the essential singularities off(z). Hence we
have a method of constructing a uniform function with any finite number of
essential singularities as a uniformly converging product of any number of
primary factors, each of which has one of the essential singularities as its sole
essential singularity and either (i) has as its sole zero either one of the zeros

t Weierstrass, I.e., p. 48.

71.] WITH ESSENTIAL SINGULARITIES 111

or one of the accidental singularities of/(V), so that it is of the form

Z — € \ a ( — .

or (ii) it has no zero and then it is of the form

/fe).

When all the primary factors of the latter form are combined, they constitute
a generalising factor in exactly the same way as in § 52 and in § 63,
Cor. III., except that now the number of essential singularities is not
limited to unity.

Two forms of expression of a function with a limited number of essential
singularities have been obtained : one (§ 65) as a sum, the other (§ 69) as a
product, of functions each of which has only one essential singularity. Inter
mediate expressions, partly product and partly sum, can be derived, e.g.
expressions of the form

z— c.

But the pure product-expression is the most general, in that it brings into
evidence not merely the n essential singularities but also the zeros and the
accidental singularities, whereas the expression as a sum tacitly requires that
the function shall have no singularities other than the n which are essential.

Note. The formation of the various elements, the aggregate of which is the complete
representation of the function with a limited number of essential singularities, can be
carried out in the same manner as in § 34 ; each element is associated with a particular
domain, the range of the domain is limited by the nearest singularities, and the aggregate
of the singularities forms the boundary of the region of continuity.

To avoid the practical difficulty of the gradual formation of the region of continuity
by the construction of the successive domains when there is a limited number of
singularities (and also, if desirable to be considered, of branch-points), Fuchs devised
a method which simplifies the process. The basis of the method is an appropriate change
of the independent variable. The result of that change is to divide the plane of the
modified variable f into two portions, one of which, G2, is finite in area and the other of
which, Gl, occupies the rest of the plane; and the boundary, common to Gl and G2, is a
circle of finite radius, called the discriminating circle* of the function. In G2 the
modified function is holomorphic ; in G^ the function is holomorphic except at f = oo ;
and all the singularities (and the branch-points, if any) lie on the discriminating circle.

The theory is given in Fuchs's memoir " Ueber die Darstellung der Functionen com-

plexer Variabeln, ," Crelle, t. Ixxv, (1872), pp. 176 — 223. It is corrected in details

and is amplified in Crelle, t. cvi, (1890), pp. 1 — 4, and in Crelle, t. cviii, (1891),
pp. 181—192; see also Nekrassoff, Math. Ann., t. xxxviii, (1891), pp. 82—90, and
Anissimoff, Math. Ann., t. xl, (1892), pp. 145—148.

* Fuchs calls it Grenzkreis.

CHAPTER VII.

FUNCTIONS WITH UNLIMITED ESSENTIAL SINGULARITIES, AND EXPANSION

IN SERIES OF FUNCTIONS.

72. IT now remains to consider functions which have an infinite number
of essential singularities*. It will, in the first place, be assumed that the
essential singularities are isolated points, that is, that they do not form a
continuous line, however short, and that they do not constitute a continuous
area, however small, in the plane. Since their number is unlimited and
their distance from one another is finite, there must be at least one point in
the plane (it may be at z = oo ) where there is an infinite aggregate of such
points. But no special note need be taken of this fact, for the character of an
essential singularity has not yet entered into question ; the essential singu
larity at such a point would merely be of a nature different from the essential
singularity at some other point.

We take, therefore, an infinite series of quantities a1} a.2, a3,... arranged in
order of increasing moduli, and such that no two are the same : and so we
have infinity as the limit of av when v = <x> .

Let there be an associated series of uniform functions of z such that
for all values of i. the function G'i ( ) , vanishing with , has a{ as its

\Z - Of/ Z — Oi

* The results in the present chapter are founded, except where other particular references are
given, upon the researches of Mittag-Leffler and Weierstrass. The most important investigations
of Mittag-Leffler are contained in a series of short notes, constituting the memoir " Sur la th6orie
des fonctions uniformes d'une variable," Comptes Rendus, t. xciv, (1882), pp. 414, 511, 713, 781,
938, 1040, 1105, 1163, t. xcv, (1882), p. 335 ; and in a memoir " Sur la representation analytique
des fonctions monogenes uniformes," Acta Math., t. iv, (1884), pp. 1 — 79. The investigations of
Weierstrass referred to are contained in his two memoirs " Ueber einen functionentheoretischen
Satz des Herrn G. Mittag-Leffler," (1880), and " Zur Functionenlehre," (1880), both included in
the volume Abhandlungen aus der Functionenlehre, pp. 53 — 66, 67 — 101, 102 — 104. A memoir by
Hermite " Sur quelques points de la theorie des fonctions," Acta Soc. Fenn., t. xii, pp. 67 — 94,
Crelle, t. xci, (1881), pp. 54 — 78 may be consulted with great advantage.

72.]

MITTAG-LEFFLER'S THEOREM

113

sole singularity; the singularity is essential or accidental according as
GI is transcendental or algebraical. These functions can be constructed
by theorems already proved. Then we have the theorem, due to Mittag-
Le frier: — It is always possible to construct a uniform analytical function F (z),
having no singularities other than a1} a«, a,, ... and such that for each deter
minate value of v, the difference F (z)-Gv ( ) is finite for z = av and

\z av/

therefore, in the vicinity of av, is expressible in the form P (z — «„).

73. To prove Mittag-Leffler's theorem, we first form subsidiary functions

Fv (z), derived from the functions G as follows. The function Gv (—- — }

\z — aj

converges everywhere in the plane except at the point «„; hence within a
circle z < av\ it is a monogenic analytic function of z, and can therefore be
expanded in a series of positive powers of z which converges uniformly
within the circle, say

z-a

for values of z such that \z\ < av . If a,, be zero, there is evidently no
expansion.

Let e be a positive quantity less than 1, and let elf e2, e3, ... be arbitrarily
chosen positive decreasing quantities, subject to the single condition that 2e
is a converging series, say of sum A : and let e0 be a positive quantity inter
mediate between 1 and e. Let g be the greatest value of ~ f

z — a,

for

points on or within the circumference \z\ = e0 a,|; then, because the series

00

2 v^z* is a converging series, we have, by § 29,

or

Hence, with values of z satisfying the condition \z\^.e av\, we have, for
any value of m,

/j.=m

Vu Z

2, q -

9 mJt

n = m fco

1-

since e<e0. Take the smallest integral value of m such that

9

F.

114 MITTAG-LEFFLER'S

it will be finite and may be denoted by mv : and thus we have

[73.

for values of z satisfying the condition \z\^.e av\.

We now construct a subsidiary function Fv (z) such that, for all values of z,

then for values of UL which are ^ e aJ,

Moreover, the function 2 zv^ is finite for all finite values of z so that, if we
n=o

take

.j

-a

— i

then 6,,(^) is zero at infinity, because, when 5=00, #„(- -)is finite by

\z — civ/

hypothesis. Evidently <f>v(z) is infinite only at z = av, and its singularity is
of the same kind as that of Gt

z — a,

74. Now let c be any point in the plane, which is not one of the points
«], a2, as, ...; it is possible to choose a positive quantity p such that no one
of the points a is included within the circle

z — c

= p-

Let av be the singularity, which is the point nearest to the origin satisfy
ing the condition «„ > c \ + p ; then, for points within or on the circle, we

have

' z

as

when s has the values v, v + 1, v + 2, Introducing the subsidiary functions

Fv (z), we have, for such values of z,

and therefore

F.(z)

a finite quantity. It therefore follows that the series 2 F, (z) converges uni-

8=v

formly and unconditionally for all values of z which satisfy the condition

74.] THEOREM 115

z — c\^.p. Moreover, all the functions Fl(z), F2(z), ..., Fr_l(z] are finite for
such values of z, because their singularities lie without the circle z — c = p ;
and therefore the series

S Fr(z)

r=l

converges uniformly and unconditionally for all points z within or on the
circle \z — c =p, where p is chosen so that the circle encloses none of the
points a.

The function, represented by the series, can therefore be expanded in the
form P (z — c), in the domain of the point c.

If am denote any one of the points a1} a2, ..., and we take p' so small that
all the points, other than am, lie without the circle

I /

I * U"m — P )

then, since Fm (z) is the only one of the functions F which has a singularity
at am, the series

^{Fr(z}}-Fm(z)

converges regularly in the vicinity of a, and therefore it can be expressed in
the form P (z — am). Hence

a

the difference of Fm and Gm being absorbed into the series P to make Pj. It

GO

thus appears that the series 2 Fr (z) is a function which has infinities only

r = \

at the points a1} a2, ..., and is such that

can be expressed in the vicinity of am in the form P (z - am). Hence 2 Fr (z)
is a function of the required kind.

75. It may be remarked that the function is by no means unique. As
the positive quantities e were subjected to merely the single condition that
they form a converging series, there is the possibility of wide variation in
their choice: and a difference of choice might easily lead to a difference
in the ultimate expression of the function.

This latitude of ultimate expression is not, however, entirely unlimited.
For, suppose there are two functions F(z) and F (z\ enjoying all the assigned
properties. Then as any point c, other than a^, a2, ..., is an ordinary point for
both F (z) and F (z), it is an ordinary point for their difference : and so

F(z)-F(z) = P(z-c)

8—2

116 FUNCTIONS POSSESSING [75.

for points in the immediate vicinity of c. The points a are, however,
singularities for each of the functions : in the vicinity of such a point a*
we have

since the functions are of the required form : hence

F(z}-F(z}=P(z-ai) -P(z- ai),

or the point a; is an ordinary point for the difference of the functions. Hence
every finite point in the plane, whether an ordinary point or a singularity
for each of the functions, is an ordinary point for the difference of the
functions : and therefore that difference is a uniform integral function of z.
It thus appears that, if F (z) be a function with the required properties, then
every other function with those properties is of the form

F(z) + G(z],

where G (z) is a uniform integral function of z either transcendental or
algebraical.

The converse of this theorem is also true.

00

Moreover, the function G (z) can always be expressed in a form 2 gv(z), if

v=\

it be desirable to do so : and therefore it follows that any function with the
assigned characteristics can be expressed in the form

76. The following applications, due to Weierstrass, can be made so as to
give a new expression for functions, already considered in Chapter VI., having
z = oo as their sole essential singularity and an unlimited number of poles at
points Oi, a2, —

If the pole at af be of multiplicity mi} then (z — a$n>f(z) is regular at
the point a; and can therefore be expressed in the form

mi— 1

Hence, if we take /f (z) = 2 c^ (z — ai)~TO<+'t,

M = 0

we have f(z} =fi (z) + P (z — «;).

Now deduce from fi(z) a function Fi(z) as in | 73, and let this deduction be

effected for each of the functions /,- (z). Then we know that

is a uniform function of z having the points a1} a2, ... for poles in the proper

76.] UNLIMITED SINGULARITIES 117

multiplicity and no essential singularity except z = oo . The most general
form of the function therefore is

r=\

Hence any uniform analytical function which has no essential singularity
except at infinity can be expressed as a sum of functions each of which has only
one singularity in the finite part of the plane. The form of Fr (z) is

fr(z}-Gr(z\

where fr (z) is infinite at z = ar and Gr (z) is a properly chosen integral
function.

We pass to the case of a function having a single essential singularity at
c and at no other point and any number of accidental singularities, by taking

z' = - as in § 63. Cor. II.: and so we obtain the theorem :

z — c

Any uniform function which has only one essential singularity, which is
at c, can be expressed as a sum of uniform functions each of which has only
one singularity different from c.

Evidently the typical summative function Fr (z) for the present case is of
the form

Z —

77. The results, which have been obtained for functions possessed of
an infinitude of singularities, are valid on the supposition, stated in § 72,
that the limit of av with indefinite increase of v is infinite ; the series
ttj, «2, ••• tends to one definite limiting point which is 2=00 and, by the
substitution z' (z — c) = 1, can be made any point c in the finite part of the
plane.

Such a series, however, does not necessarily tend to one definite limiting
point: it may, for instance, tend to condensation on a curve, though the
condensation does not imply that all points of the continuous arc of the curve
must be included in the series. We shall not enter into the discussion
of the most general case, but shall consider that case in which the series of
moduli \al) a2 , ... tends to one definite limiting value so that, with in
definite increase of v, the limit of \av is finite and equal to R ; the points
«i, «2, ... tend to condense on the circle \z = R.
Such a series is given by

2fori
( I _ l -m+n

«„,*={! +

for &=0, 1, ..., n, and n=l, 2, ... ad inf.; and another* by

a«Hl + (-l)ncn}e2M7"V2,
where c is a positive proper fraction.

* The first of these examples is given by Mittag-Leffler, Acta Math., t. iv, p. 11 ; the second
was stated to me by Mr Burnside.

118 FUNCTIONS POSSESSING [77.

With each point am we associate the point on the circumference of the
circle, say bm, to which am is nearest: let

| dm "m I = Pm>

so that pm approaches the limit zero with indefinite increase of m. There
cannot be an infinitude of points ap, such that pp^<&, any assigned positive
quantity ; for then either there would be an infinitude of points a within or
on the circle \z\ = R — ®, or there would be an infinitude of points a within
or on the circle z = R + ©, both of which are contrary to the hypothesis
that, with indefinite increase of v, the limit of \av is R. Hence it follows
that a finite integer n exists for every assigned positive quantity ®, such that

\am-bm\ < ®
when m^n.

Then the theorem, which corresponds to Mittag-LefHer's as stated in § 72
and which also is due to him, is as follows : —

It is always possible to construct a uniform analytical function of z which
exists over the whole plane, except at the points a and b, and which, in the
immediate vicinity of each one of the singularities a, can be expressed in the form

where the functions G{ are assigned functions, vanishing with - - and finite

Z — (Li

everywhere in the plane except at the single points a; with which they are
respectively associated.

In establishing this theorem, we shall need a positive quantity e less than
unity and a converging series e^ e2, e3, ... of positive quantities, all less than
unity.

Let the expression of the function Gn be

"I / _. .. \0 I / - _. \5 ' ' ' ' '

n \z - a ~ z-an (z- an)2 (z - an)s

Then, since z - an = (z - bn) \l -- — ~l\ ,

( z on )

the function Gn can be expressed* in the form

l«— <li

for values of z such that

an -

z-bn
and the coefficients A are given by the equations

* The justification of this statement is to be found in the proposition in § 82.

77.]

UNLIMITED SINGULARITIES

119

Now, because Gn is finite everywhere in the plane except at an, the series

has a finite value, say #, for any non-zero value of the positive quantity %n ;
then

Hence

0*-!)!

ft & f

< S flfr-^

71 ?^

Introducing a positive quantity a such that

we choose £n so that £n < a|an - bn\ ;

and then | A n> ^ \ < go. ( 1 + a)*-1.

Because (1 + a) e is less than unity, a quantity 6 exists such that

(1 + a) e < 6 < 1.

Then for values of z determined by the condition

go. 6

dn on

< e, we have

al-0'

Let the integer mn be chosen so that

ga &>

it will be a finite integer, because 0< 1. Then

00 (1 7)

V I A I "H ^^

We now construct, as in § 73, a subsidiary function Fn(z), defining it by
the equation

so that for points z determined by the condition

\Fn(z)\<en.
A function with the required properties is

00

Fm(z\

< €, we have

m=l

120 FUNCTIONS POSSESSING [77.

To prove it, let c be any point in the plane distinct from any of the points
a and b ; we can always find a value of p such that the circle

\z-c\=p

contains none of the points a and b. Let I be the shortest distance between
this circle and the circle of radius R, on which all the points b lie ; then for

all points z within or on the circle

z — c

— p we have

Now we have seen that, for any assigned positive quantity <s), there is a
finite integer n such that

I dm — bm < ©

when m ^ n. Taking ® = el, we have

m

< e

when m^n,n being the finite integer associated with the positive quantity el.
It therefore follows that, for points z within or on the circle \z — c\ = p,

\Fm(z}\<em,
when m is not less than the finite integer n. Hence

a finite quantity because e1} e2, ... is a converging series; and therefore

is a converging series. Each of the functions F1(z), F»(z), ..., Fn_-i(z) is
finite when z — c ^ p ; and therefore

is a series which converges uniformly and unconditionally for all values of z
included in the region

\z-c\^p.

Hence the function represented by the series can be expressed in the form
P (z — c) for all such values of z. The function therefore exists over the
whole plane except at the points a and b.

It may be proved, exactly as in § 74, that, for points z in the immediate
vicinity of a singularity am,

The theorem is thus completely established.

The function thus obtained is not unique, for a wide variation of choice of
the converging series ea + e2 + . . . is possible. But, in the same way as in the

77.]

UNLIMITED SINGULARITIES

121

corresponding case in § 75, it is proved that, if F (z) be a function with the
required properties, every other function with those properties is of the form

F(z}+G(z\

where G (z) behaves regularly in the immediate vicinity of every point in the
plane except the points b.

78. The theorem just given regards the function in the light of an
infinite converging series of functions of the variable : it is natural to suppose
that a corresponding theorem holds when the function is expressed as an
infinite converging product. With the same series of singularities as in
§ 77, when the limit of av with indefinite increase of v is finite and
equal to R, the theorem* is: —

It is always possible to construct a uniform analytical function which
behaves regularly everywhere in the plane except at the points a and b and
which in the vicinity of any one of the points av can be expressed in the
form

where the numbers w1} n2, ... are any assigned integers.

The proof is similar in details to proofs of other propositions and it will
therefore be given only in outline. We have

au-

provided

such values of z,

z-av z-bv z - bv ^i V z - bv J '
< e, the notation being the same as in § 77. Hence, for

=e

(/7 _ 7) \
i_ ^ _ M
2-bJ

-n,, S

by Ev (z), we have Ev (z} =e m"

Hence, if F(z) denote the infinite product

we have F(z) = e

and F(z) is a determinate function provided the double series in the index of

the exponential converge.

* Mittag-Leffler, Acta Math., t. iv, p. 32 ; it may be compared with Weierstrass's theorem in
§67.

122 TRANSCENDENTAL FUNCTION AS

Because nv is a finite integer and because

[78.

is a converging series, it is possible to choose an integer mv so that

7)

"x

M(T^

where t]v is any assigned positive quantity. We take a converging series of
positive quantities rjv : and then the moduli of the terms in the double series
form a converging series. The double series itself therefore converges
uniformly and unconditionally ; and then the infinite product F (z) converges
uniformly and unconditionally for points z such that

&„ — b..

< e.

As in § 77, let c be any point in the plane, distinct from any of the
points a and b. We take a finite value of p such that the circle z — c\=p
contains none of the points a and b ; and then, for all points within or on this
circle,

z—

<e

when m^n, n being the finite integer associated with the positive quantity
el. The product

fi Ev(z)

v=n

is therefore finite, for its modulus is less than

CO

S IJK

K = »

the product

n

v=l

is finite, because the circle z — c\ = p contains none of the points a and 6;
and therefore the function F(z) is finite for all points within or on the circle.
Hence in the vicinity of c, the function can be expanded in the form P (z — c) ;
and therefore the function exists everywhere in the plane except at the points
a and b.

The infinite product converges ; it can be zero only at points which make
one of the factors zero and, from the form of the factors, this can take place
only at the points av with positive integers nv. In the vicinity of av all
the factors of F (z) except Ev (z) are regular ; hence F (z)\Ev (z) can be
expressed as a function of z — av in the vicinity. But the function has no
zeros there, and therefore the form of the function is

Pl (z-a,,).

78.] AN INFINITE SERIES OF FUNCTIONS 123

Hence in the vicinity of av, we have

on combining with Pl (z — av) the exponential index in Ev(z). This is the
required property.

Other general theorems will be found in Mittag-Leffler's memoir just
quoted.

79. The investigations in §§ 72 — 75 have led to the construction of a
function with assigned properties. It is important to be able to change, into
the chosen form, the expression of a given function, having an infinite series
of singularities tending to a definite limiting point, say to z = oo . It is
necessary for this purpose to determine (i) the functions Fr(z) so that the

00

series 2 Fr (z) may converge uniformly and (ii) the function G (z).

r=l

Let <& (z) be the given function, and let S be a simple contour embracing
the origin and /j, of the singularities, viz., al , ...... , aM: then, if t be any

point, we have

- « « .

m r« *£) ,,y r« *y) ,,.

J t-z\t) J t-z\t)

f(a) _

where I implies an integral taken round a very small circle centre a.

If the origin be one of the points a1} a2, ...... , then the first term will be

included in the summation.

Assuming that z is neither the origin nor any one of the points a1} ..., a^,
we have

so

27TI

AT ^

Now — . 7-^-7 dt

1 [(0)$>(t)fz\
— . 7-^-7
Ziri] t-z\tj

,
— -—. 2 I 7—^- - dt.

t-Z\t)

(ffl-l)i I

\~dm 1(®(t) + ^i^+^

[

124 TRANSCENDENTAL FUNCTION AS [79.

unless z = 0 be a singularity and then there will be no term G (z). Similarly,
it can be shewn that

/ I \ m-l / z \ A

is equal to Gv(- -} - 2 vj-} = F, (z),

\z - aj A=0 \aj

where , — s— •

2?rt

and the subtractive sum of m terms is the sum of the first m terms in the
development of Gv in ascending powers of z. Hence

If, for an infinitely large contour, m can be chosen so that the integral

t-

diminishes indefinitely with increasing contours enclosing successive singu
larities, then

The integer m may be called the critical integer.
If the origin be a singularity, we take

and there is then no term G (z) : hence, including the origin in the summa
tion, we then have

so that if, for this case also, there be some finite value of m which makes
the integral vanish, then

Other expressions can be obtained by choosing for m a value greater than
the critical integer ; but it is usually most advantageous to take m equal to
its least lawful value.

Ex. 1. The singularities of the function ?r cot 772 are given by z = \, for all integer
values of X from — oo to +00 including zero, so that the origin is a singularity.

The integral to be considered is

- 1 M IT cot vt fz\m ,,
= ~ — . I — - (- ) at.

2iri J t-z \tj

We take the contour to be a circle of very large radius R chosen so that the circumference
does not pass innuitesimally near any one of the singularities of TT coint at infinity; this

79.] AN INFINITE SERIES OF FUNCTIONS 125

is, of course, possible because there is a finite distance between any two of them. Then,
round the circumference so taken, n cot nt is never infinite : hence its modulus is never
greater than some finite quantity M.

Let t = Reei, so that ~=id6; then

v

and therefore

Z

.--—.
t-z

for some point t on the circle. Now, as the circle is very large, we have \t-z\ infinite :
hence \J\ can be made zero merely by taking m unity.

Thus, for the function TT cot TTZ, the critical integer is unity.
Hence from the general theorem we have the equation

1 fir cot nt z j

7T COt 772= -5— . 2 I— -dt,

2TTI J t-Z t

the summation extending to all the points X for integer values of X = - oc to + oo , and
each integral being taken round a small circle centre X.

-vr . » . 1 /"(*) TT cot irt z ,

Now if, in -— . • -dt.

2m J t - z t

we take t=\ + (, we have

where P(Q = 0 when £= 0; and therefore the value of the integral is

•*./ (*-*+{) (x+fl t

In the limit when |f| is infinitesimal, this integral

z

= (X-2)X

1 1

~X-2 X'

and therefore /*. (z) = -J— + 1

A ' z-X X'

if X be not zero.

And for the zero of X, the value of the integral is

(p

126 REGION OF CONTINUITY [79.

so that F0(z) is -. In fact, in the notation of § 72, we have

z

o P-A»JL

^ \z-\J~z-\'
arid the expansion of GK needs to be carried only to one term.

1 A=ao /I 1\
We thus have 7rcot7rs = — f- 2 — N+=r)>

z A=-co \Z-X A/

the summation not including the zero value of X.
Ex. 2. Obtain, ab initio, the relation

SHI2 3 A=_aj (z-X7r)2'

p. 3. Shew that, if

1 °° 1 1

then "-^^ = - + 2z 2 ^3-^1-

R(z) z i=lR(\)z*-\*

(Gylden, Mittag-Leffler.)

Ex. 4. Obtain an expression, in the form of a sum, for

IT cot irz

where Q(z) denotes (1 -z) (l -^ (l -|J ...... ^-j)*-

80. The results obtained in the present chapter relating to functions
which have an unlimited number of singularities, whether distributed over
the whole plane or distributed over only a finite portion of it, shew that
analytical functions can be represented, not merely as infinite converging
series of powers of the variable, but also as infinite converging series of
functions of the variable. The properties of functions when represented by
series of powers of the variable depended in their proof on the condition that
the series proceeded in powers; and it is therefore necessary at least to
revise those properties in the case of functions when represented as series
of functions of the variable.

Let there be a series of uniform functions /i (z), /, (z), . . . ; then the
aggregate of values of z, for which the series

1*1

has a finite value, is the region of continuity of the series. If a positive
quantity p can be determined such that, for all points z within the circle

z — a\ = p,

80.] OF A SERIES OF FUNCTIONS 127

00

the series 2 fi(z) converges uniformly and unconditionally*, the series is

said to converge in the vicinity of a. If R be the greatest value of p for
which this holds, then the area within the circle

z — a\ = R

is called the domain of a; and the series converges uniformly and uncon
ditionally in the vicinity of any point in the domain of a.

It will be proved in § 82 that the function can be represented by power-
series, each such series being equivalent to the function within the domain of
some one point. In order to be able to obtain all the power-series, it is
necessary to distribute the region of continuity of the function into domains
of points where it has a uniform, finite value. We therefore form the domain
of a point 6 in the domain of a from a knowledge of the singularities of the
function, then the domain of a point c in the domain of 6, and so on ; the
aggregate of these domains is a continuous part of the plane which has
isolated points and which has one or several lines for its boundaries. Let
this part be denoted by At.

For most of the functions, which have already been considered, the region
A1} thus obtained, is the complete region of continuity. But examples will
be adduced almost immediately to shew that A-^ does not necessarily include
all the region of continuity of the series under consideration. Let a' be a
point not in A-^ within whose vicinity the function has a uniform, finite
value ; then a second portion A2 can be separated from the whole plane, by
proceeding from a' as before from a. The limits of A± and A2 may be wholly
or partially the same, or may be independent of one another : but no point
within either can belong to the other. If there be points in the region of con
tinuity which belong to neither A1 nor A2, then there must be at least another
part of the plane A3 with properties similar to At and^l2- And so on. The

00

series 2 fi(z) converges uniformly and unconditionally in the vicinity of

»=i «

every point in each of the separate portions of its region of continuity.

It was proved that a function represented by a series of powers has a
definite finite derivative at every point lying actually within the circle
of convergence of the series, but that this result cannot be affirmed for a
point on the boundary of the circle of convergence even though the value of
the series itself should be finite at the point, an illustration being provided
by the hypergeometric series at a point on the circumference of its circle of

* In connection with most of the investigations in the remainder of this chapter, Weierstrass's
memoir " Zur Functionenlehre " already quoted (p. 112, note) should be consulted.

It may be convenient to give here Weierstrass's definition (I.e., p. 70) of uniform, unconditional

convergence. A series 2 fn converges uniformly, if an integer m can be determined so that

/»

can be made less than any arbitrary positive quantity, however small ; and it converges uncon
ditionally, if the uniform convergence of the series be independent of any special arrangement
of order or combination of the terms.

128 REGION OF CONTINUITY OF [80.

convergence. It will appear that a function represented by a series of
functions has a definite finite derivative at every point lying actually within
its region of continuity, but that the result cannot be affirmed for a point
on the boundary; and an example will be given (§ 83) in which the derivative
is indefinite.

Again, it has been seen that a function, initially defined by a given power-
series, is, in most cases, represented by different analytical expressions in
different parts of the plane, each of the elements being a valid expression of
the function within a certain region. The questions arise whether a given
analytical expression, either a series of powers or a series of functions :
(i) can represent different functions in the same continuous part of its region
of continuity, (ii) can represent different functions in distinct (that is, non-
continuous) parts of its region of continuity.

81. Consider first a function defined by a given series of powers.

Let there be a region A' in the plane and let the region of continuity of
the function, say g (z), have parts common with A'. Then if a0 be any point
in one of these common parts, we can express g {z) in the form P (z — a0) in
the domain of a0.

As already explained, the function can be continued from the domain of
a0 by a series of elements, so that the whole region of continuity is gradually
covered by domains of successive points ; to find the value in the domain of
any point a, it is sufficient to know any one element, say, the element in the
domain of a0. The function is the same through its region of continuity.

Two distinct cases may occur in the continuations.

First, it may happen that the region of continuity of the function g (z)
extends beyond A'. Then we can obtain elements for points outside A',
their aggregate being a uniform analytical function. The aggregate of
elements then represents within A' a single analytical function : but as that
function has elements for points without A, the aggregate within A' does
not completely represent the function. Hence

If a function be defined within a continuous region of a plane by an
aggregate of elements in the form of power-series, which are continuations of
one another, the aggregate represents in that part of the plane one (and only
one) analytical function : but if the power-series can be continued beyond the
boundary of the region, the aggregate of elements within the region is not the
complete representation of the analytical function.

This is the more common case, so that examples need not be given.

Secondly, it may happen that the region of continuity of the function does
not extend beyond A' in any direction. There are then no elements of the
function for points outside A' and the function cannot be continued beyond
the boundary of A. The aggregate of elements is then the complete
representation of the function and therefore :

81.] A SERIES OF POWERS 129

If a function be defined within a continuous region of a plane by an
aggregate of elements in the form of power-series, which are continuations of
one another, and if the power-series cannot be continued across the boundary of
that region, the aggregate of elements in the region is the complete representa
tion of a single uniform monogenic function which exists only for values of the
variable within the region.

The boundary of the region of continuity of the function is, in the latter
case, called the natural limit of the function*, as it is a line beyond which
the function cannot be continued. Such a line arises for the series

l + 2z + ^ + 2z9 + ... ,

in the circle \z = 1, a remark due to Kronecker; other illustrations occur in
connection with the modular functions, the axis of real variables being the
natural limit, and in connection with the automorphic functions (see Chapter
XXII.) when the fundamental circle is the natural limit. A few examples
will be given at the end of the present Chapter.

It appears that Weierstrass was the first to announce the existence of natural limits
for analytic functions, Berlin Monatsber. (1866), p. 617 ; see also Schwarz, Ges. Werke,
t. ii, pp. 240 — 242, who adduces other illustrations and gives some references ; Klein and
Fricke, Vorl. uber die Theorie der elliptischen Modulfunctioncn, t. i, (1890), p. 110; Jordan,
Cows d' Analyse, t. iii, pp. 609, 610. Some interesting examples and discussions of
functions, which have the axis of real variables for a natural limit, are given by Hankel,
" Untersuchungen liber die unendlich oft oscillirenden und unstetigen Functionen,"
Math. Ann., t. xx, (1870), pp. 63—112.

82. Consider next a series of functions of the variable ; let it be

The region of continuity may be supposed to consist of several distinct parts,
in the most general case ; let one of them be denoted by A. Take some
point in A, say the origin, which is either an ordinary point or an isolated
singularity; and let two concentric circles of radii R and R' be drawn in A,
so that

R < z =r<R,

and the space between these circles lies within A. In this space, each term
of the series is finite and the whole series converges uniformly and uncon
ditionally.

Now let fi (z) be expanded in a series of powers of z, which series con
verges within the space assigned, and in that expansion let ^ be the co-

oo

efficient of z* ; then we can prove that 2 i^ is finite and that the series

( / °° \

s |(sO

n. (\i = 0 I

* Die natiirliche Grenze, according to German mathematicians.
F.

130 REGION OF CONTINUITY [82.

converges uniformly and unconditionally within this space, so that

•x. (/ oo

2 /,(*) = 2 2

i=l " /A {\i=Q

00

Because the infinite series 2 fi (z) converges uniformly and uncon
ditionally, a number n can be chosen so that

where & is an arbitrary finite quantity, ultimately made infinitesimal; and
therefore also

i=n

where n' > n and is infinite in the limit. Now since the number of terms in
the series

is not infinite before the limit, we have

But the original series converges unconditionally, and therefore k is not less

n

than the greatest value of the modulus of 2 fi(z) for points within the

i=n

region; hence, by § 29, we have

n

2 V < AT <i.
»•=»

00

Moreover, A; is not less than the greatest value of the modulus of 2 fi(z)
in the given region ; and so

00

2 i^ < AT *.

i=n

Now, by definition, k can be made as small as we desire by choice of n ; hence

the series

is a converging series. Let it be denoted by A^.

n-l oo

Let 2 r'M = A /, 2 ifj, = A M" ;

then, by the above suppositions, we can always choose n so that

k being any assignable small quantity.

82.]

OF A SERIES OF FUNCTIONS

131

When two new quantities r± and r2 are introduced, as in § 28, satisfying
the inequalities

f-f ^ ly ^ \ iv --• /y» ^ 7?'

-il/<^/l<s.|.S|<i./2<.-fl,

the integer w can be chosen so that

\Ap'\ < kr~* < kr^.

f- r.

Then

and

so that

2

.—

00

2

-

- <k

M=-oo r — r-i r2-r

Hence the series 2 A^'z^ can by choice of n be made to have a
modulus less than any finite quantity ; and therefore, since

/u.= oo n — 1

(for there is a finite number of terms in the coefficients on each side, the
expansions are converging series, and the sum on the right-hand side is a
finite quantity), it follows that the series

converges uniformly.
Finally, we have

2 .

fl= —00

2 ft (*) - 24^ = 2 /< (z) -
<=i 1=1

and therefore

2

t'=n

r ~

which, as k can be diminished indefinitely, can be made less than any finite

jlX=00

quantity. Hence the series 2 A^ converges unconditionally, and there-

fi= —00

fore we have

provided

00 jlt=00

2 /;(*)= 2 .

l'=l /u= — oo

9—2

132 REGION OF CONTINUITY [82.

When we take into account all the parts of the region of continuity
of the series, constituted by the sum of the functions, we have similar
expansions in the form of successive series of powers of z — c, converging
uniformly and unconditionally in the vicinities of the successive points c.
But, in forming the domains of these points c, the boundary of the region of
continuity of the function must not be crossed ; and a new series of powers is
required when the circle of convergence of any one series (lying within the
region of continuity) is crossed.

It therefore appears that a converging series of functions of a variable
can be expressed in the form of series of powers of the variable which
converge within the parts of the plane where the series of functions
converges uniformly and unconditionally ; but the equivalence of the two
expressions is limited to such parts of the plane and cannot be extended
beyond the boundary of the region of continuity of the series of functions.

If the region of continuity of a series of functions consist of several parts
of the plane, then the series of functions can in each part be expressed in
the form of a set of converging series of powers : but the sets of series of
powers are not necessarily the same for the different parts, and they are not
necessarily continuations of one another, regarded as power-series.

Suppose, then, that the region of continuity of a series of functions

F(z)=lfi(z)

i=l

consists of several parts A1} A.2, Within the part A^ let F (z) be

represented, as above, by a set of power-series. At every point within A1}
the values of F(z) and of its derivatives are each definite and unique ; so
that, at every point which lies in the regions of convergence of two of the
power-series, the values which the two power-series, as the equivalents of F (z)
in their respective regions, furnish for F (z) and for its derivatives must be
the same. Hence the various power-series, which are the equivalents of F (z)
in the region Aly are continuations of one another: and they are sufficient to
determine a uniform monogenic analytic function, say F^ (z}. The functions
F(z) and Fl(z) are equivalent in the region Al; and therefore, by § 81, the
series of functions represents one and the same function for all points within
one continuous part of its region of continuity. It may (and frequently does)
happen that the region of continuity of the analytical function F± (z) extends
beyond A± ; and then F-^ (z) can be continued beyond the boundary of A^ by
a succession of elements. Or it may happen that the region of continuity
of Fl (z) is completely bounded by the boundary of A^ ; and then the function
cannot be continued across that boundary. In either case, the equivalence

00

of F-L(Z) and 2 fi(z) does not extend beyond the boundary of Alt one

82.] OF A SERIES OF FUNCTIONS 133

00

complete and distinct part of the region of continuity of 2 fi(z); and

i = \

therefore, by using the theorem proved in § 81, it follows that :

A series of functions of a variable, which converges within a continuous part
of the plane of the variable z, is either a partial or a complete representation
of a single uniform, analytic function of the variable in that part of the plane.

83. Further, it has just been proved that the converging series of
functions can, in any of the regions A, be changed into an equivalent
uniform, analytic function, the equivalence being valid for all points in
that region, say

2 /(•). 4(4

i = l

But for any point within A, the function Fl(z) has a uniform finite derivative

oo

(§ 21); and therefore also 2 fi(z) has a uniform finite derivative. The

i=l

equivalence of the analytic function and the series of functions has not been
proved for points on the boundary; even if they are equivalent there, the
function I\ (z) cannot be proved to have a uniform finite derivative at every

00

point on the boundary of A, and therefore it cannot be affirmed that 2 ft (z)

i=\

has, of necessity, a uniform, finite derivative at points on the boundary of A, even

oo

though the value of 2 fi(z) be uniform and finite at every point on the

i=l

boundary*.

Ex. In illustration of the inference just obtained, regarding the derivative of a
function at a point on the boundary of its region of continuity, consider the series

g(z)= 2 &V",

n=0

where b is a positive quantity less than unity, and a is a positive quantity which will be
taken to be an odd integer.

For points within and on the circumference of the circle \z =1, the series converges
uniformly and unconditionally; and for all points without the circle the series diverges.
It thus defines a function for points within the circle and on the circumference, but not
for points without the circle.

Moreover for points actually within the circle the function has a first derivative and
consequently has any number of derivatives. But it cannot be declared to have a
derivative for points on the circle: and it will in fact now be proved that, if a certain
condition be satisfied, the derivative for variations at any point on the circle is not merely
infinite but that the sign of the infinite value depends upon the direction of the variation,
so that the function is not monogenic for the circumference t.

* It should be remarked here, as at the end of § 21, that the result in itself does not contravene
Biemann's definition of a function, according to which (§ 8) -^ must have the same value what
ever be tbe direction of the vanishing quantity dz ; at a point on the boundary of the region
there are outward directions for which die is not defined.

t The following investigation is due to Weierstrass, who communicated it to Du Bois-Eeymond :
see Crclle, t. Ixxix, (1875), pp. 29—31.

134

A SERIES OF FUNCTIONS

[83.

Let z = eei: then, as the function converges unconditionally for all points along the
circle, we take

f(ff)= 2 lnea"ei,

71=0

where 6 is a real variable. Hence

m-l IV,an(0 + 4>)*_,,«WWl

= s«nH - — -
H=O 1 an$ J

/•ea">+»> (0 + <f>) i _ ea™+«0(S

+ 2 &w + M - -T - 1 1

«=o I 9 J

assuming m, in the first place, to be any positive integer. To transform the first sum on
the right-hand side, we take

and therefore

pan (0 + <j>) i _ a"0i

2 (ab}n

n=0

<M21^n 8Jn(fr-*)

if ab>\. Hence, on this hypothesis, we have

2 (ab)n \ — \ =y r i »

*=o ( a"0 J ao - 1

where 7 is a complex quantity with modulus <1.

To transform the second sum on the right-hand side, let the integer nearest to am
be am, so that

7T

for any value of m : then taking

we have \tr^-x> — %n,

and cos x is not negative. We choose the quantity <f> so that

and therefore

TT am

ff) — — — ,
0

which, by taking m sufficiently large (a is > 1), can be made as small as we please. We
now have

am+"(6 +<i>)i = Qaniti (1 + o™) _ _ / _ j N°™

if a be an odd integer, and

_

am+nOi _ ani (x + iram] _ / _ j \<»meana;i
, a"xi

Hence

CD /•

and therefore 2 &- + « f

,,=0 i

_

- ( - 1)

2 6"

83.] MAY NOT POSSESS A DERIVATIVE 135

The real part of the series on the right-hand side is

2 bn{l + cosanx};

n=0

every term of this is positive and therefore, as the first term is 1 + cos x, the real part

> 1+cos.r

>1
for cos x is not negative ; and it is finite, for it is

<2 2 bn

K=0

2

<r^6-

Moreover far < TT — x < frr,

so that -- is positive and >-. Hence
TT — x 6

where TJ is a finite complex quantity, the real part of which is positive and greater than
unity. We thus have

where |y'|<l, and the real part of 77 is positive and > 1.
Proceeding in the same way and taking

IT ' am '

TT+X

so that % = — — ,

we find — — — — t_LJ — _ ( _ iy™ (a^

where |y/|<l and the real part of TJ^ a finite complex quantity, is positive and greater
than unity.

If now we take ab - 1 > fn-,

the real parts of - — + y -*-—= , say of f,

O 7T (tO — 1

and of |li+yi'__L_,sayof fl,

are both positive and different from zero. Then, since

and ~x- = (_!)«- (ab)m d ,

/(.

m being at present any positive integer, we have the right-hand sides essentially different
quantities, because the real part of the first is of sign opposite to the real part of the second.

Now let m be indefinitely increased; then $ and x are infinitesimal quantities
which ultimately vanish ; and the limit of - [/(# + </>)-/(#)] for $ = 0 is a complex infinite

136 ANALYTICAL EXPRESSION [83.

quantity with its real part opposite in sign to the real part of the complex infinite quantity
which is the limit of $ — ~^ f°r = ®- If# had a differential coefficient

A

these two limits would be equal : hence / (0) has not, for any value of 6, a determinate
differential coefficient.

From this result, a remarkable result relating to real functions may be at once derived.
The real part of / (<9) is

2 6ncos(an<9),

n=0

which is a series converging uniformly and unconditionally. The real parts of

-(-ir («&)-<:

and of +(-l)am(a6)TOf1

are the corresponding magnitudes for the series of real quantities : and they are of opposite
signs. Hence for no value of 6 has the series

2 6"cos(an<9)

n=0

a determinate differential coefficient, that is, we can choose an increase <£ and a decrease ^
of 6, both being made as small as we please and ultimately zero, such that the limits of
the expressions

0 -X

are different from one another, provided a be an odd integer and ab > 1 +|TT.

The chief interest of the above investigation lies in its application to functions of real
variables, continuity in the value of which is thus shewn not necessarily to imply the
existence of a determinate differential coefficient defined in the ordinary way. The
application is due to Weierstrass, as has already been stated. Further discussions will
be found in a paper by Wiener, Crelle, t. xc, (1881), pp. 221 — 252, in a remark by
Weierstrass, Abh. aus der Functionenlehre, (1886), p. 100, and in a paper by Lerch, Crelle,
t. ciii, (1888), pp. 126 — 138, who constructs other examples of continuous functions of
real variables ; and an example of a continuous function without a derivative is given by
Schwarz, Ges. Werke, t. ii, pp. 269 — 274.

The simplest classes of ordinary functions are characterised by the properties : —
(i) Within some region of the plane of the variable they are uniform, finite and

continuous :
(ii) At all points within that region (but not necessarily on its boundary) they have

a differential coefficient :

(iii) When the variable is real, the number of maximum values and the number of
minimum values within any given range is finite.

The function 2 bn cos (anQ\ suggested by Weierstrass, possesses the first but not the

71=0

second of these properties. Kb'pcke (Math. Ann., t. xxix, pp. 123 — 140) gives an example
of a function which possesses the first and the second but not the third of these
properties.

84. In each of the distinct portions Alt A.2>... of the complete region of
continuity of a series of functions, the series can be represented by a
monogenic analytic function, the elements of which are converging power-
series. But the equivalence of the function -series and the monogenic

84.] REPRESENTING DIFFERENT FUNCTIONS 137

analytic function for any portion A^ is limited to that region. When the
monogenic analytic function can be continued from A^ into Az, the continua
tion is not necessarily the same as the monogenic analytic function which is

00

the equivalent of the series 2 fi(z) in A2. Hence, if the monogenic analytic

i = l

functions for the two portions A^ and A2 be different, the function-series
represents different functions in the distinct parts of its region of continuity.

A simple example will be an effective indication of the actual existence
of such variety of representation in particular cases ; that, which follows, is
due to Tannery*.

Let a, b, c be any three constants ; then the fraction

a + bczm
Y+'bzm '
when m is infinite, is equal to a if z \ < 1, and is equal to c if | z > 1.

Let m0, m1} m2>... be any set of positive integers arranged in ascending
order and be such that the limit of mn, when n = oo , is infinite. Then,

since

a + bczm* a + bczm° » {a + bczmi a + bczm

1 + bzm» 1 -f bzm° f.i (1 + bzmi I + bz'"
^mo

" ~* a)

the function <f)(z), defined by the equation

,. a + bczm° ., N S f 0^-^-1-1)^-1

+ (z} = TT6^ + b (G ~ a) £ {(I + bz^) (i + 6^

converges uniformly and unconditionally to a value a if \ z < 1, awe? converges
uniformly and unconditionally to a value c if z \ > 1. But it does not con
verge uniformly and unconditionally if z \ = 1.

The simplest case occurs when b = — 1 and m^ = 2* ; then, denoting the
function by <f> (z), we have

a - cz , . ( z z2 z4

that is, the function <f> (z) is equal to a if z < 1, and it is equal to c if

* It is contained in a letter of Tannery's to Weierstrass, who communicated it to the Berlin
Academy in 1881, Abh. aus der Functionenlehre, pp. 103, 104. A similar series, which indeed is
equivalent to the special form of $ (z), was given by Schroder, Schlfim. Zeitschrift, t. xxii, (1876),
p. 184; and Pringsheim, Math. Ann., t. xxii, (1883), p. 110, remarks that it can be deduced,
without material modifications, from an expression given by Seidel, Crelle, t. Ixxiii, (1871),
pp. 297- -299.

138 LINE OF SINGULARITIES [84.

When \z =\, the function can have any value whatever. Hence a circle
of radius unity is a line of singularities, that is, it is a line of discontinuity
for the series. The circle evidently has the property of dividing the plane
into two parts such that the analytical expression represents different
functions in the two parts.

If we introduce a new variable £ connected with z by the relation*

l +z

then, if £= £ + iy and z = x + iy, we have

1 rfS. nil

fc i — x y

so that £ is positive when \z\< 1, and £ is negative when \ z \ > 1. If then

the function %(£) is equal to a or to c according as the real part of f is
positive or negative.

And, generally, if we take £ a rational function of z and denote the
modified form of </> (£), which will be a sum of rational functions of z, by
^(z), then <f>i(z) will be equal to a in some parts of the plane and to c
in other parts of the plane. The boundaries between these parts are lines
of singular points : and they are constituted by the ^-curves which correspond
to £| = 1.

85. Now let F(z) and G(z) be two functions of z with any number of
singularities in the plane : it is possible to construct a function which shall
be equal to F (z} within a circle centre the origin and to G (z) without the
circle, the circumference being a line of singularities. For, when we make
a = 1 and c = 0 in </> (z) of § 84, the function

1 z z* z4

00)=- -- + -. — r + •-: — : + -. -^r + . . .
V/ 1—0 Z2 — I Z*— I ZS —I

is unity for all points within the circle and is zero for all points without it :

and therefore

G(z} + {F(z)-G(z)}6(z}

is a function which has the required property.

Similarly F3 (z) + {F, (z) - F, (z)} 6 (z) + {F, (z) - F3 (z}} 6 (

is a function which has the value Fl (z) within a circle of radius unity, the value F2 (z)
between a circle of radius unity and a concentric circle of radius r greater than unity, and
the value F3(z) without the latter circle. All the singularities of the functions F1} F2, F3
are singularities of the function thus represented; and it has, in addition to these, the
two lines of singularities given by the circles.

* The significance of a relation of this form will be discussed in Chapter XIX.

85.] MONOGENIC FUNCTIONALITY 139

Again, 6

is a function of s, which is equal to F(z) on the positive side of the axis of y, and is equal
to G (z) on the negative side of that axis.

1+2

Also, if we take £e l —p\ = ^~i

where ax and p1 are real constants, as an equation defining a new variable £ + iy, we have
| cos at + 77 sin aj -pl = p. \23T~2

so that the two regions of the 2-plane determined by \z\<l and \z\>l correspond to the
two regions of the {"-plane into which the line £ cos a: + 77 sin al—p1 = 0 divides it. Let

,-«'ai — », — 1\

so that on the positive side of the line £ cos at + 77 sin aj — p1 = 0 the function 6l is unity and
on the negative side of that line it is zero. Take any three lines defined by ax, p1; a2, p2',
a,, pn respectively ; then

AJ.A11 (2)\-F/(l)

is a function which has the value F within
the triangle, the value - F in three of the
spaces without it, and the value zero in the
remaining three spaces without it, as indi
cated in the figure (fig. 13).

And for every division of the plane by
lines, into which a circle can be transformed (3)
by rational equations, as will be explained
when conformal representation is discussed (1) /

hereafter, there is a possibility of represent- Fig. 13.

ing discontinuous functions, by expressions similar to those just given.

These examples are sufficient to lead to the following result*, which is
complementary to the theorem of § 82 :

When the region of continuity of an infinite series of functions consists
of several distinct parts, the series represents a single function in each part
but it does not necessarily represent the same function in different parts.

It thus appears that an analytical expression of given form, which con
verges uniformly and unconditionally in different parts of the plane separated
from one another, can represent different functions of the variable in those
different parts ; and hence the idea of monogenic functionality of a complex
variable is not coextensive with the idea of functional dependence expressible
through arithmetical operations, a distinction first established by Weierstrass.

86. We have seen that an analytic function has not a definite value at
an essential singularity and that, therefore, every essential singularity is
excluded from the region of definition of the function.

* Weierstrass, I.e., p. 90.

140 SINGULAR LINES [86.

Again, it has appeared that not merely must single points be on occasion
excluded from the region of definition but also that functions exist with
continuous lines of essential singularities which must therefore be excluded.
One method for the construction of such functions has just been indicated :
but it is possible to obtain other analytical expressions for functions which
possess what may be called a singular line. Thus let a function have a
circle of radius c as a line of essential singularity*; let it have no other
singularities in the plane and let its zeros be al} a2, a3,..., supposed arranged
in such order that, if pneie" = an> then

I Pn C | ^ Pn+i ~ C >

so that the limit of pn, when n is infinite, is c.

Let cn = ceie«, a point on the singular circle, corresponding to an which is
assumed not to lie on it. Then, proceeding as in Weierstrass's theory in § 51,
if

«.= oo („ _

Gz= n

where gn(z) = - + L_ +... + _

Z-Cn 2 \Z-CnJ mn - I \ Z - Cn

G (z) is a uniform function, continuous everywhere in the plane except along
the circumference of the circle which may be a line of essential singularities.

Special simpler forms can be derived according to the character of the
series of quantities constituted by | an - cn . If there be a finite integer m,

00

such that 2 an — cn m is a converging series, then in gn (z) only the first

M = l

m — 1 terms need be retained.
Ex. Construct the function when

m being a given positive integer and r a positive quantity.

Again, the point cn was associated with an so that they have the same
argument : but this distribution of points on the circle is not necessary and
can be made in any manner which satisfies the condition that in the limited

00

case just quoted the series 2 an — cn m is a converging series.

Singular lines of other classes, for example, sectioiis\ in connection with functions
defined by integrals, arise in connection with analytical functions. They are discussed
by Painleve, "Sur les lignes singulieres des fonctions analytiques," (These, Gauthier-
Villars, Paris, 1887).

Ex. Shew that, if the zeros of a function be the points

. _b+c— (a — d) i

ZT ^ ~7 i 7T \ • 5

* This investigation is due to Picard, Comptes Rendus, t. xci, (1881), pp. 690—692.
t Called conpures by Hermite ; see § 103.

86.] LACUNARY FUNCTIONS 141

where a, ?;, c, d are integers satisfying the condition ad-bo = l, so that the function
has a circle of radius unity for an essential singular line, then if

b + di

„

2J = -^ - =— , ,

d+bi'

( A

the function n \ 5 e z

(z — li

where the product extends to all positive integers subject to the foregoing condition
ad-bc = l, is a uniform function finite for all points in the plane not lying on the
circle of radius unity. (Picard.)

87. In the earlier examples, instances were given of functions which
have only isolated points for their essential singularities : and, in the later
examples, instances have been given of functions which have lines of
essential singularities, that is, there are continuous lines for which the
functions do not exist. We now proceed to shew how functions can be
constructed which do not exist in assigned continuous spaces in the plane,
these spaces being aggregates of essential singularities. Weierstrass was
the first to draw attention to lacunary functions, as they may be called ;
the following investigation in illustration of Weierstrass's theorem is due to
Poincare' *.

Take any convex curve in the plane, say G ; and consider the function

*z^b'
where the quantities A are constants, subject to the conditions

(i) The series ^\A\ converges uniformly and unconditionally :
(ii) Each of the points b is either within or on the curve G :
(iii) The points b are the aggregate of all rational j points within and
on C : then the function is a uniform analytical function for all points
without C and it has the area of G for a lacunary space.

First, it is evident that, if z = b, then the series does not converge.
Moreover as the points b are the aggregate of all the rational points within
or on C, there will be an infinite number of singularities in the immediate
vicinity of b : we shall thus have an unlimited number of terms each infinite
of the first order, and thus (§ 42) the point b will be an essential singularity.
As this is true of all points z within or on C, it follows that the area C is a
lacunary space for the function, if the function exist at all.

Secondly, let z be a point without G ; and let d be the distance of z from
the nearest point of the boundary of C^f% so that d is not a vanishing quantity.

* Acta Soc. Fenn., t. xii, (1883), pp. 341—350.

J Rational points within or on C are points whose positions can be determined rationally in
terms of the coordinates of assigned points on C ; examples will be given.

t This will be either the shortest normal from z to the boundary or the distance of z from
some point of abrupt change of direction, as for instance at the angular point of a polygon.

142

FUNCTIONS WITH

[87.

Then | z — b \ ^ d ; and therefore

A _ \A\ \A\
~\z-b\< d '

z-b

so that

-b

A

z-b

Now 2 j.A| converges uniformly and unconditionally and therefore, as d does
not vanish,

z-b
converges uniformly and unconditionally, that is.

is a function of 2 which converges uniformly and unconditionally for every
point without C. Let it be denoted by <£ (z).

Let c be any point without C, and let r be the radius of the greatest
circle centre c which can be drawn so as to have no point of C within itself
or on its circumference, so that r is the radius of the domain of c; then
b — c > r, for all points b.

If we take a point z within this circle, we have \z — c =6r, where 6 < 1.

Now for all points within this circle the function <£ {z} converges uniformly,

A

and every term -- =• of <f> (z) is finite. Also, for points within the circle, we

A

can expand -- j in powers of z — c in the form

of a converging series. Hence, by § 82, we have

<£(*)= 2 Bm(z-c)m,

a series converging uniformly and unconditionally for all points within the
circle centre c and radius r, which circle is the circle of convergence of the
series. The function can be expressed in the usual manner over the whole of
the region of continuity, which is the part of the plane without the curve C.

Thus 0 (z) is a uniform analytical function, having the area of C for a
lacunary space.

As an example, take a convex polygon having o1} ...... , ap for its angular points;

then any point

...... +mj>ap

TOI + ...... +mp

where mlt ...... , mp are positive integers or zero (simultaneous zeros being excluded), is

87.] LACUNARY SPACES 143

either within the polygon or on its boundary : and any rational point within the polygon

or on its boundary can be represented by

p
2 mrar

r=l

P '
2 mr
r=l
by proper choice of ?n15 ...... , mp, a choice which can be made in an infinite number of ways.

Let ult ...... , Up be given quantities, the modulus of each of which is less than unity:

then the series

•9-11 m> 11 mf

«& ^ ' I ...... ftp

o

converges uniformly and unconditionally. Then all the assigned conditions are satisfied
for the function

_ .. . + mpap > '

ml + ...... +mp J

and therefore it is a function which converges uniformly and unconditionally everywhere
outside the polygon and which has the polygonal space (including the boundary) for
a lacunary space.

If, in particular, p = Z, we obtain a function which has the straight line
joining ax and a2 as a line of essential singularity. When we take at = 0,
a.2 = 1 and slightly modify the summation, we obtain the function

2 2 ^ 2

w=l m=0 W&

7i

which, when u^ <\ and |w2|<l, converges uniformly and unconditionally
everywhere in the plane except at points between 0 and 1 on the axis of real
quantities, this part of the axis being a line of essential singularity.
For the general case, the following remarks may be made :

(i) The quantities u1} u2>... need not be the same for every term; a
numerator, quite different in form, might be chosen, such as
(mj2+ ... + m/)"'1 where 2//, > p ; all that is requisite is that the
series, made up of the numerators, should converge uniformly
and unconditionally.

(ii) The preceding is only a particular illustration and is not necessarily
the most general form of function having the assigned lacunary
space.

It is evident that the first step in the construction of a function, which
shall have any assigned lacunary space, is the formation of some expression
which, by the variation of the constants it contains, can be made to
represent indefinitely nearly any point within or on the contour of the
space. Thus for the space between two concentric circles of radii a and c
and centre the origin we should take

Wja + O-WjU ^a«
-a£- e n

n

144 EXAMPLES [87.

which, by giving m^ all values from 0 to n, ra2 all values from 0 to n — 1 and
n all values from 1 to infinity will represent all rational points in the space :
and a function, having the space between the circles as lacunary, would be
given by

oo n n-1

2 2 2

n=l »»!=(> m2=0

(n — raj) b ^ 271-

•r /3

.6 — C

n

provided u\ < 1, u^ < 1, u2 < 1.

In particular, if a = 6, then the common circumference is a line of essential singularity
for the corresponding function. It is easy to see that the function

z — ae n

ao 2n-l m n

provided the series 2 2 u v

n=l m=0 m,n m, n

converges uniformly and unconditionally, is a function having the circle |0| = a as a line of
essential singularity.

Other examples will be found in memoirs by Goursat*, Poincaref, and HomenJ.

Ex. 1. Shew that the function

where r is a real positive quantity and the summation is for all integers m and n between
the positive and the negative infinities, is a uniform function in all parts of the plane
except the axis of real quantities which is a line of essential singularity.

Ex. 2. Discuss the region in which the function

w=i m=i jf/=i i ^- . ••- •
2—1 1 i

\7i 71

is definite. (Homen.)

Ex. 3. Prove that the function

n=0

exists only within a circle of radius unity and centre the origin. (Poincare.)

Ex. 4. An infinite number of points at, a2, as, are taken on the circumference of

a given circle, centre the origin, so that they form the aggregate of rational points on the
circumference. Shew that the series

2 l Z

can be expanded in a series of ascending powers of z which converges for points within the
circle, but that the function cannot be continued across the circumference of the circle.

(Stieltjes.)

* Comptes Rendus, t. xciv, (1882), pp. 715—718 ; Bulletin de Darboux, 2me Ser. , t. xi, (1887),
pp. 109—114.

t In the memoir, quoted p. 138, and Comptes Rendus, t. xcvi, (1883), pp. 1134-1136.
+ Acta Soc. Fenn., t. xii, (1883), pp. 445—464.

87.] EXAMPLES 145

Ex. 5. Prove that the series

2 | :

7T .00 -" K1-2TO-
9 oo

22

~li)2) '

7T _oo _oo (^(1 — 2wi — 2nz i) \zm-\-Anz~

where the summation extends over all positive and negative integral values of ra and of n
except simultaneous zeros, is a function which converges uniformly and unconditionally
for all points in the finite part of plane which do not lie on the axis of y ; and that
it has the value +1 or - 1 according as the real part of z is positive or negative.

(Weierstrass.)
Ex. 6. Prove that the region of continuity of the series

consists of two parts, separated by the circle z\ = l which is a line of infinities for
the series : and that, in these two parts of the plane, it represents two different
functions.

_<a'ir

If two complex quantities a> and to' be taken, such that z = e ^ and the real part of
^. is positive, and if they be associated with the elliptic function $ (u) as its half-periods,
then for values of z which lie within the circle z = \

in the usual notation of Weierstrass's theory of elliptic functions.

Find the function which the series represents for values of z without the circle \z\ = \.

(Weierstrass.)

Ex. 7. Four circles are drawn each of radius -^ having their centres at the points

1, i, - 1, -i respectively; the two parts of the plane, excluded by the four circumferences,
are denoted the interior and the exterior parts. Shew that the function

n='K sini^TT ( 1 1 1 1

is equal to IT in the interior part and is zero in the exterior part. (Appell.)

Ex. 8. Obtain the values of the function

»;-l- (-!)•(, i >1 l

«=i n V1 •> (2 + l)« (2-l)«J

in the two parts of the area within a circle centre the origin and radius 2 which lie

without two circles of radius unity, having their centres at the points 1 and - 1

respectively. (Appell.)

Ex- 9- If
and

,~3 ......

amr (2-«m)3 J

where the regions of continuity of the functions F extend over the whole plane, then / (z)
is a function existing everywhere except within the circles of radius unity described round
the points a, , «2, ...... , an. (Teixeira.)

F- 10

146 CLASSIFICATION [87.

Ex. 10. Let there be n circles having the origin for a common centre, and let
£,, (72, ...... , (7n, C'n + 1 be % + 1 arbitrary constants; also let a1} a2, ...... , an be any w points

lying respectively on the circumferences of the first, the second, ...... , the nth circles.

Shew that the expression

1 ("(CL
27T./0 W*

has the value <7m for points z lying between the (»w - l)th and the with circles and the
value (7n + 1 for points lying without the nth circle.

Construct a function which shall have any assigned values in the various bands into
which the plane is divided by the circles. (Pincherle.)

88. In § 32 it was remarked that the discrimination of the various
species of essential singularities could be effected by means of the properties
of the function in the immediate vicinity of the point.

Now it was proved, in § 63, that in the vicinity of an isolated essential
singularity b the function could be represented by an expression of the form

for all points in the space without a circle centre b of small radius and within
a concentric circle of radius not large enough to include singularities at
a finite distance from b. Because the essential singularity at b is isolated,
the radius of the inner circle can be diminished to be all but infinitesimal :

the series P (z — b) is then unimportant compared with G I —31 ) , which
can be regarded as characteristic for the singularity of the function.

Another method of obtaining a function, which is characteristic of the
singularity, is provided by § 68. It was there proved that, in the vicinity of
an essential singularity a, the function could be represented by an expression
of the form

where, within a circle of centre a and radius not sufficiently large to include
the nearest singularity at a finite distance from a, the function Q (z — a) is
finite and has no zeros : all the zeros of the given function within this circle
(except such as are absorbed into the essential singularity at a) are zeros of

the factor H ( - - ] , and the integer-index n is affected by the number of these
zeros. When the circle is made small, the function

z-a

can be regarded as characteristic of the immediate vicinity of a or, more
briefly, as characteristic of a.

88.] OF SINGULARITIES 147

It is easily seen that the two characteristic functions are distinct. For
if F and F^ be two functions, which have essential singularities at a of the
same kind as determined by the first characteristic, then

F(z)-Fl(z) = P(z-a)-Pl(z-a)
= P,(z-a\

while if their singularities at a be of the same kind as determined by the
second characteristic, then

F(z)_Q(z-a)
f\(*)-Q^-~a) = Q^2-

in the immediate vicinity of a, since Q1 has no zeros. Two such equations
cannot subsist simultaneously, except in one instance.

Without entering into detailed discussion, the results obtained in the
preceding chapters are sufficient to lead to an indication of the classification
of singularities*.

Singularities are said to be of the first class when they are accidental ;
and a function is said to be of the first class when all its singularities are of
the first class. It can, by § 48, have only a finite number of such singularities,
each singularity being isolated.

It is for this case alone that the two characteristic functions are in
accord.

When a function, otherwise of the first class, fails to satisfy the last
condition, solely owing to failure of finiteness of multiplicity at some point,
say at z = x , then that point ceases to be an accidental singularity. It has
been called (§ 32) an essential singularity ; it belongs to the simplest kind of
essential singularity ; and it is called a singularity of the second class.

A function is said to be of the second class when it has some singularities
of the second class ; it may possess singularities of the first class. By an
argument similar to that adopted in § 48, a function of the second class
can have only a limited number of singularities of the second class, each
singularity being isolated.

When a function, otherwise of the second class, fails to satisfy the last
condition solely owing to unlimited condensation at some point, say at z = oo ,
of singularities of the second class, that point ceases to be a singularity
of the second class: it is called a singularity (necessarily essential) of the
third class.

* For a detailed discussion, reference should be made to Guichard, " Theorie des points
singnhers essentiels" (These, Gauthier-ViUars, Paris, 1883), who gives adequate references to the

:stigations of Mittag-Leffler in the introduction of the classification and to the researches of
Cantor. See also Mittag-Leffler, Acta Math., t. iv, (1884), pp. 1_79; Cantor Crelle t Ixxxiv
1878), pp. 242—258, Acta Math., t. ii, (1883), pp. 311—328.

10—2

148 CLASSIFICATION OF SINGULARITIES [88.

A function is said to be of the third class when it has some singularities
of the third class ; it may possess singularities of the first and the second
classes. But it can have only a limited number of singularities of the third
class, each singularity being isolated.

Proceeding in this gradual sequence, we obtain an unlimited number of
classes of singularities: and functions of the various classes can be constructed
by means of the theorems which have been proved. A function of class n
has a limited number of singularities of class n, each singularity being
isolated, and any number of singularities of lower classes which, except in so
far as they are absorbed in the singularities of class n, are isolated points.

The effective limit of this sequence of classes is attained when the
number of the class increases beyond any integer, however large. When
once such a limit is attained, we have functions with essential singularities of
unlimited class, each singularity being isolated ; when we pass to functions
which have their essential singularities no longer isolated but, as in previous
class-developments, of infinite condensation, it is necessary to add to the
arrangement in classes an arrangement in a wider group, say, in species*.

Calling, then, all the preceding classes of functions functions of the first
species, we may, after Guichard (I.e.), construct, by the theorems already
proved, a function which has at the points al} a*,... singularities of classes
1, 2,..., both series being continued to infinity. Such a function is called
a function of the second species.

By a combination of classes in species, this arrangement can be continued
indefinitely ; each species will contain an infinitely increasing number of
classes; and when an unlimited number of species is ultimately obtained,
another wider group must be introduced.

This gradual construction, relative to essential singularities, can be carried
out without limit ; the singularities are the characteristics of the functions.

* Guichard (I.e.) uses the term genre.

CHAPTER VIII.

MULTIFORM FUNCTIONS.

89. HAVING now discussed some of the more important general properties
of uniform functions, we proceed to discuss some of the properties of multiform
functions.

Deviations from uniformity in character may arise through various causes :
the most common is the existence of those points in the ^-plane, which have
already (§ 12) been defined as branch-points.

As an example, consider the two power-series

Wl = l-i/-i/2-... , W2 = _(i_i/_^_... )f

which, for points in the plane such that z' is less than unity, are the two
values of (1 - /)* ; they may be regarded as two branches of the function w
defined by the equation

w2 = 1 — z' = z.

Let / describe a small curve (say a circle of radius r) round the point
z' = l, beginning on the axis of x\ the point 1 is the origin for z. Then z
is r initially, and at the end of the first description of the circle z is re2wi ;
hence initially wl is + 14 and w.2 is - r*} and at the end of the description
w1 is -f r^e™ and w2 is — r^e™, that is, wl is — rf and w.2 is + ri Thus the
effect of the single circuit is to change wl into w.2 and w2 into w1} that is,
the effect of a circuit round the point, at which w1 and w2 coincide in value,
is to interchange the values of the two branches.

If, however, z describe a circuit which does not include the branch-point,
wl and w2 return each to its initial value.

Instances have already occurred, e.g. integrals of uniform functions, in
which a variation in the path of the variable has made a difference in the

150 CONTINUATIONS [89.

result; but this interchange of value is distinct from any of the effects
produced by points belonging to the families of critical points which have
been considered. The critical point is of a new nature ; it is, in fact, a
characteristic of multiform functions at certain associated points.

We now proceed to indicate more generally the character of the relation
of such points to functions affected by them.

The method of constructing a monogenic analytic function, described in
§ 34, by forming all the continuations of a power-series, regarded as a given
initial element of the function, leads to the aggregate of the elements of the
function and determines its region of continuity. When the process of con
tinuation has been completely carried out, two distinct cases may occur.

In the first case, the function is such that any and every path, leading
from one point a to another point z by the construction of a series of
successive domains of points along the path, gives a single value at z as the
continuation of one initial value at a. When, therefore, there is only a
single value of the function at a, the process of continuation leads to only a
single value of the function at any other point in the plane. The function is
uniform throughout its region of continuity. The detailed properties of such
functions have been considered in the preceding chapters.

In the second case, the function is such that different paths, leading from
a to z, do not give a single value at z as the continuation of one and the
same initial value at a. There are different sets of elements of the function,
associated with different sets of consecutive domains of points on paths from
a to z, which lead to different values of the function at z; but any change
in a path from a to z does not necessarily cause a change in the value of the
function at z. The function is multiform in its region of continuity. The
detailed properties of such functions will now be considered.

90. In order that the process of continuation may be completely carried
out, continuations must be effected, beginning at the domain of any point a
and proceeding to the domain of any other point b by all possible paths in
the region of continuity, and they must be effected for all points a and b.
Continuations must be effected, beginning in the domain of every point a
and returning to that domain by all possible closed paths in the region of
continuity. When they are effected from the domain of one point a to that
of another point b, all the values at any point z in the domain of a (and not
merely a single value at such points) must be continued : and similarly when
they are effected, beginning in the domain of a and returning to that domain.
The complete region of the plane will then be obtained in which the function
can be represented by a series of positive integral powers : and the boundary
of that region will be indicated.

90.] OF A MULTIFORM FUNCTION 151

In the first instance, let the boundary of the region be constituted by a
number, either finite or infinite, of
isolated points, say L1} L2, Ls, ...
Take any point A in the region, so
that its distance from any of the
points L is not infinitesimal ; and
in the region draw a closed path
ABC...EFA so as to enclose one
point, say Ll} but only one point, of
the boundary and to have no point

of the curve at a merely infinitesimal distance from L^ Let such curves be
drawn, beginning and ending at A, so that each of them encloses one and
only one of the points of the boundary : and let Kr be the curve which
encloses the point Lr.

Let Wj be one of the power-series defining the function in a domain with
its centre at A : let this series be continued along each of the curves Ks by
successive domains of points along the curve returning to A. The result
of the description of all the curves will be that the series w^ cannot be
reproduced at A for all the curves though it may be reproduced for some
of them ; otherwise, w: would be a uniform function. Suppose that w.2, w3> ...,
each in the form of a power-series, are the aggregate of new distinct values
thus obtained at A ; let the same process be effected on w2, w3, ... as has
been effected on w1; and let it further be effected on any new distinct values
obtained at A through w2, w3, ... , and so on. When the process has
been carried out so far that all values obtained at A, by continuing any
series round any of the curves K back to A, are included in values already
obtained, the aggregate of the values of the function at A is complete : they
are the values at A of the branches of the function.

We shall now assume that the number of values thus obtained is finite,
say n, so that the function has n branches at A : if their values be denoted
by w1} w2, ..., wn, these n quantities are all the values of the function at A.
Moreover, n is the same for all points in the plane, as may be seen by con
tinuing the series at A to any other point and taking account of the corollaries
at the end of the present section.

The boundary-points L may be of two kinds. It may (and not infre
quently does) happen that a point Ls is such that, whatever branch is taken
at A as the initial value for the description of the circuit Ks, that branch is
reproduced at the end of the circuit. Let the aggregate of such points be
/u J2, .... Then each of the remaining points L is such that a description
of the circuit round it effects a change on at least one of the branches, taken
as an initial value for the description ; let the aggregate of these points be
Blt 52, .... They are the branch-points; their association with the definition
in § 12 will be made later.

152

DEFORMATION OF PATH

[90.

Fig. 15.

When account is taken of the continuations of the function from a point
A to another point B, we have n values at B as the continuations of n values
at A. The selection of the individual branch at B, which is the continuation
of a particular branch at A, depends upon the path of z between A and 5;
it is governed by the following fundamental proposition : —

The final value of a branch of a function for two paths of variation of the
independent variable from one point to another will be the same, if one path
can be deformed into the other without passing over a branch-point.

Let the initial and the final points be a and b, and let one path of
variation be acb. Let another path of variation be aeb,
both paths lying in the region in which the function can
be expressed by series of positive integral powers : the two
paths are assumed to have no point within an infinitesimal
distance of any of the boundary-points L and to be taken
so close together that the circles of convergence of pairs of
points (such as cx and e1} c2 and e2, and so on) along the two
paths have common areas. When we begin at a with a
branch of the function, values at d and at e^ are obtained,
depending upon the values of the branch and its derivatives at a and upon
the positions of ca and e^ hence, at any point in the area common to the
circles of convergence of these two points, only a single value arises as
derived through the initial value at a. Proceeding in this way, only a single
value is obtained at any point in an area common to the circles of con
vergence of points in the two paths. Hence ultimately one and the same
value will be obtained at b as the continuation of the value of the one branch
at a by the two different paths of variation which have been taken so that
no boundary-point L lies between them or infinitesimally near to them.

Now consider any two paths from a to b, say acb and adb, such that
neither of them is near a boundary-point and that the
contour they constitute does not enclose a boundary-point.
Then by a series of successive infinitesimal deformations we
can change the path acb to adb ; and as at b the same value
of w is obtained for variations of z from a to b along the
successive deformations, it follows that the same value of w
is obtained at b for variations of z along acb as for varia
tions along adb.

Next, let there be two paths acb, adb constituting a closed contour,
enclosing one (but not more than one) of the points / and none of the points
B. When the original curve K which contains the point / is described, the
initial value is restored : and hence the branches of the function obtained at
any point of K by the two paths from any point, taken as initial point, are
the same. By what precedes, the parts of this curve K can be deformed

Fig. 16.

90.] OF THE VARIABLE 153

into the parts of acbda without affecting the branches of the function : hence
the value obtained at b, by continuation along acb, is the same as the value
there obtained by continuation along adb. It therefore follows that a path
between two points a and b can be deformed over any point / without
affecting the value of the function at b ; so that, when the preceding
results are combined, the proposition enunciated is proved.

By the continued application of the theorem, we are led to the following
results : —

COROLLARY I. Whatever be the effect of the description of a circuit on the
initial value of a function, a reversal of the circuit restores the original value
of the function.

For the circuit, when described positively and negatively, may be re
garded as the contour of an area of infinitesimal breadth, which encloses no
branch-point within itself and the description of the contour of which
therefore restores the initial value of the function.

COROLLARY II. A circuit can be deformed into any other circuit without
affecting the final value of the function, provided that no branch-point be crossed
in tJie process of deformation.

It is thus justifiable, and it is often convenient, to deform a path con
taining a single branch-point into a loop round the
point. A loop* consists of a line nearly to the point, °~
nearly the whole of a very small circle round the point, Fig. 17.

and a line back to the initial point; see figure 17.

COROLLARY III. The value of a function is unchanged when the variable
describes a closed circuit containing no branch-point ; it is likewise unchanged
when the variable describes a closed circuit containing all the branch-points.

The first part is at once proved by remarking that, without altering the
value of the function, the circuit can be deformed into a point.

For the second part, the simplest plan is to represent the variable on
Neumann's sphere. The circuit is then a curve on the sphere enclosing all
the branch-points : the effect on the value of the function is unaltered by any
deformation of this curve which does make it cross a branch-point. The
curve can, without crossing a branch-point, be deformed into a point in that
other part of the area of the sphere which contains none of the branch
points ; and the point, which is the limit of the curve, is not a branch
point. At such a point, the value of the function is unaltered ; and there
fore the description of a circuit, which encloses all the branch-points,
restores the initial value of the function.

COROLLARY IV. If the values of w at b for variations along two paths

* French writers use the word lacet, German writers the word Schleife.

154

EFFECT OF DEFORMATION

[90.

acb, adb be not the same, then a description of acbda will not restore the initial
value of w at a.

In particular, let the path be the loop OeceO (fig. 17), and let it change w
at 0 into w'. Since the values of w at 0 are different and because there is
no branch-point in Oe (or in the evanescent circuit OeO), the values of w at
e cannot be the same : that is, the value with which the infinitesimal circle
round a begins to be described is changed by the description of that circle.
Hence the part of the loop that is effective for the change in the value of w is
the small circle round the point ; and it is because the description of a small
circle changes the value of w that the value of w is changed at 0 after the
description of a loop.

If/0?) be the value of w which is changed mtof^z) by the description of
the loop, so that/Oz) and f^(z) are the values at 0, then the foregoing
explanation shews that /(e) and / (e) are the values at e, the branch /(e)
being changed by the description of the circle into the branch /i(e).

From this result the inference can be derived that the points Bl} B.2, ...
are branch-points as defined in § 12. Let a be any one of the points, and
let f(z) be the value of w which is changed into f, (z) by the description of
a very small circle round a. Then as the branch of w is monogenic, the
difference between f(z) and f^(z) is an infinitesimal quantity of the same
order as the length of the circumference of the circle : so that, as the circle
is infinitesimal and ultimately evanescent, \f(z) -/iOz)| can be made as small
as we please with decrease of z - a or, in the limit, the values of /(a) and
/(a) at the branch-point are equal. Hence each of the points B is such
that two or more branches of the function have the same value at the point
and there is interchange among these branches when the variable describes a
small circuit round the point : which affords a definition of a branch-point,
more complete than that given in § 12.

COROLLARY V. If a closed circuit contain several branch-points, the effect
which it produces can be obtained by a combination of the effects produced in
succession by a set of loops each going round only one of the branch-points.

If the circuit contain several branch-points, say three as at a, b, c, then a
path such as AEFD, in fig. 18, can without
crossing any branch-point, be deformed into the
loops AaB, BbC, GcD; and therefore the complete
circuit AEFD A can be deformed validly into
AaBbCcDA, and the same effect will be produced
by the two forms of circuit. When D is made DA

practically to coincide with A, the whole of the Fig. 18.

second circuit is composed of the three loops. Hence the corollary.

This corollary is of especial importance in the consideration of integrals
of multiform functions.

91.] OF PATH OF THE VAKIABLE 155

COROLLARY VI. In a continuous part of the plane where there are no
branch-points, each branch of a multiform function is uniform.

Each branch is monogenic and, except at isolated points, continuous;
hence, in such regions of the plane, all the propositions which have been
proved for monogenic analytic functions can be applied to each of the
branches of a multiform function.

91. If there be a branch-point within the circuit, then the value of the
function at 6 consequent on variations along acb may, but will not necessarily,
differ from its value at the same point consequent on variations along adb.
Should the values be different, then the description of the whole curve acbda
will lead at a not to the initial value of w, but to a different value.
The test as to whether such a change is effected by the description is
immediately derivable from the foregoing proposition; and as in Corollary
IV., § 90, it is proved that the value is or is not changed by the loop,
according as the value of w for a point near the circle of the loop is
or is not changed by the description of that circle. Hence it follows that, if
there be a branch-point which affects the branch of the function, a path of
variation of the independent variable cannot be deformed across the branch
point without a change in the value of w at the extremity of the path.

And it is evident that a point can be regarded as a branch-point for a
function only if a circuit round the point interchange some (or all) of the
branches of the function which are equal at the point. It is not necessary that
all the branches of the function should be thus affected by the point : it is
sufficient that some should be interchanged*.

Further, the change in the value of w for a single description of a circuit
enclosing a branch-point is unique.

For, if a circuit could change w into w' or w", then, beginning with w"
and describing it in the negative sense we should return to w and afterwards
describing it in the positive sense with w as the initial value we should
obtain w'. Hence the circuit, described and then reversed, does not restore
the original value w" but gives a different branch w' ; and no point on
the circuit is a branch-point. This result is in opposition to Corollary I.,
of § 90 ; and therefore the hypothesis of alternative values at the end of
the circuit is not valid, that is, the change for a single description is
unique.

But repetitions of the circuit may, of course, give different values at the
end of successive descriptions.

* In what precedes, certain points were considered which were regular singularities (see
p. 163, note) and certain which were branch-points. Frequently points will occur which are at
once branch-points and infinities ; proper account must of course be taken of them.

156

LAW OF INTERCHANGE

[92.

Fig. 19.

92. Let 0 be any ordinary point of the function ; join it to all the
branch-points (generally assumed finite in
number) in succession by lines which do not
meet each other : then each branch is uniform
for each path of variation of the variable which
meets none of these lines. The effects pro
duced by the various branch-points and their
relations on the various branches can be indi
cated by describing curves, each of which
begins at a point indefinitely near 0 and
returns to another point indefinitely near it
after passing round one of the branch- points,
and by noting the value of each branch of the function after each of these
curves has been described.

The law of interchange of branches of a function after description of a
circuit round a branch-point is as follows: —

All the branches of a function, which are affected by a branch-point as such,
can either be arranged so that the order of interchange (for description of a
path round the point) is cyclical, or be divided into sets in each of which the
order of interchange is cyclical.

Let wlt w.2> w3)... be the branches of a function for values of z near a
branch-point a which are affected by the description of a small closed curve
C round a : they are not necessarily all the branches of the function, but only
those affected by the branch-point.

The branch w^ is changed after a description of C ; let w2 be the branch
into which it is changed. Then w2 cannot be unchanged by C; for a reversed
description of C, which ought to restore w1} would otherwise leave w.2 un
changed. Hence w2 is changed after a description of (7; it may be changed
either into w1 or into a new branch, say w3. If into wlt then w-^ and w2 form
a cyclical set.

If the change be into w3, then w3 cannot remain unchanged after a
description of C, for reasons similar to those that before applied to the
change of w.2: and it cannot be changed into w2, for then a reversed de
scription of G would change wz into w.A, and it ought to change w2 into w^
Hence, after a description of C, w3 is changed either into w^ or into a new
branch, say w4. If into w1} then w1} w2, w3 form a cyclical set.

If the change be into w4, then w4 cannot remain unchanged after a
description of G ; and it cannot be changed into w.2 or ws, for by a reversal
of the circuit that earlier branch would be changed into w4 whereas it ought
to be changed into the branch, which gave rise to it by the forward descrip
tion — a branch which is not w4. Hence, after a description of C, w4 is
changed either into w^ or into a new branch. If into wlf then wj} w.2, w3, w4
form a cyclical set.

92.] OF BRANCHES OF A FUNCTION 157

If w4 be changed into a new branch, we proceed as before with that new
branch and either complete a cyclical set or add one more to the set. By
repetition of the process, we complete a cyclical set sooner or later.

If all the branches be included, then evidently their complete system
taken in the order in which they come in the foregoing investigation is a
system in which the interchange is cyclical.

If only some of the branches be included, the remark applies to the set
constituted by them. We then begin with one of the branches not included
in that set and evidently not inclusible in it, and proceed as at first, until
we complete another set which may include all the remaining branches or
only some of them. In the latter case, we begin again with a new branch
and repeat the process ; and so on, until ultimately all the branches are
included. The whole system is then arranged in sets, in each of which the
order of interchange is cyclical.

93. The analytical test of a branch-point is easily obtained by con
structing the general expression for the branches of a function which are
interchanged there.

Let z = a be a branch-point where n branches w1} ^v2,..., wn are cyclically
interchanged. Since by a first description of a small curve round a, the
branch w1 changes into w2, the branch w» into ws, and so on, it follows that
by r descriptions w1 is changed into wr+l and by n descriptions wl reverts to
its initial value. Similarly for each of the branches. Hence each branch
returns to its initial value after n descriptions of a circuit round a branch
point where n branches of the function are interchangeable.

Now let z - a = Zn ;

then, when z describes circles round a, Z moves in a circular arc round its
origin. For each circumference described by z, the variable Z describes

-th part of its circumference; and the complete circle is described by Z
round its origin when n complete circles are described by z round a. Now
the substitution changes wr as a function of z into a function of Z, say into
Wr; and, after n complete descriptions of the ^-circle round a, wr returns
to its initial value. Hence, after the description of a ^-circle round its
origin, Wr returns to its initial value, that is, Z = 0 ceases to be a branch
point for Wr. Similarly for all the branches W.

But no other condition has been associated with a as a point for the
function w ; and therefore Z = 0 may be any point for the function W, that
is, it may be an ordinary point, or a singularity. In every case we have W
a uniform function of Z in the immediate vicinity of the origin ; and therefore
in that vicinity it can be expressed in the form

158 ANALYTICAL EXPRESSION [93.

with the significations of P and G already adopted. When Z is an ordinary
point, G is a constant or zero ; when Z is an accidental singularity, O is an
algebraical function ; and, when Z is an essential singularity, G is a transcen
dental function.

The simpler cases are, of course, those in which the form of G is alge
braical or constant or zero ; and then W can be put into the form

ZmP(Z),

where P is an infinite series of positive powers and m is an integer. As this
is the form of W in the vicinity of Z=Q, it follows that the form of w in the
vicinity of z = a is

m 1

(z - a)n P {(z - a)n}

and the various n branches of the function are easily seen to be given by

i
substituting in the above for (z — a)n the values

2im j.

e m (z — of,

where s = 0, 1,..., n — 1. We therefore infer that the general expression for
the n branches of a function, which are interchanged by circuits round a
branch-point z = a, assumed not to be an essential singularity, is

m _ 1

(z - a)Tl P {(z - a)»},

i

where m is an integer, and where to (z — a)n its n values are in turn assigned
to obtain the different branches of the function.

There may be, however, more than one cyclical set of branches. If there
be another set of r branches, then it may similarly be proved that their

general expression is

OTI _ i

(zjaYQ{(z-ay-},

where m^ is an integer, and Q is an integral function ; the various branches

i
are obtained by assigning to (z — a)r its r values in turn.

And so on, for each of the sets, the members of which are cyclically
interchangeable at the branch-point.

When the branch-point is at infinity, a different form is obtained. Thus
in the case of a set of n cyclically interchangeable branches we take

z = %-»,

so that n negative descriptions of a closed £-curve, excluding infinity and no
other branch-point, requires a single positive description of a closed curve
round the w-origin. These n descriptions restore the value of w; as a function
of z to its initial value; and therefore the single description of the M- curve
round the origin restores the value of U — the equivalent of w after the

93.] NEAR A BRANCH-POINT 159

change of the independent variable — as a function of u. Thus u = 0 ceases
to be a branch-point for the function U ; and therefore the form of U is

. .

where the symbols have the same general signification as before.

If, in particular, z = oo be a branch-point but not an essential singularity,
then G is either a constant or an algebraical function ; and then U can be
expressed in the form

u~mP(u},

where TO is an integer. When the variable is changed from u to z, then the
general expression for the n branches of a function which are interchangeable

at z = oo , assumed not to be an essential singularity, is

where TO is an integer and where to zn its n values are assigned to obtain the
different branches of the function.

If, however, the branch-point z = a in the former case or z = oo in the
latter be an essential singularity, the forms of the expressions in the vicinity
of the point are

_i i

G{(z-a) »J>-p{(jr-aJ»}f

i _i

and G(zn) + P(z n),

respectively.

Note. When a multiform function is denned, either explicitly or im
plicitly, it is practically always necessary to consider the relations of the
branches of the function for z = oo as well as their relations for points that
are infinities of the function. The former can be determined by either
of the processes suggested in § 4 for dealing with z=<x>; the latter can be
determined as in the present article.

Moreover, the total number of branches of the function has been assumed
to be finite. The cases, in which the number of branches is unlimited, need
not be discussed in general : it will be sufficient to consider them when they
arise, as they do arise, e.g., when the function is of the form of an algebraical
irrational with an irrational index such as z^ — hardly a function in the
ordinary sense—, or when the function is the logarithm of a function of z,
or is the inverse of a periodic function. In the nature of their multiplicity
of branching and of their sequence of interchange, they are for the most part
distinct from the multiform functions with only a finite number of branches.

Ex. The simplest illustrations of multiform functions are furnished by functions
denned by algebraical equations, in particular, by algebraic irrationals.

160

ALGEBRAICAL

[93.

The general type of the algebraical irrational is the product of a number of functions
of the form w = {A(z — al)(z-a.2) ...... (z-a^)}m, m and n being integers.

This particular function has m branches; the points a1} «2, ...... , an are branch-points.

To find the law of interchange, we take z-ar = pe01; then when a small circle of radius p
is described round ar, so that z returns to its initial position, the value of 6 increases by

2n and the new value of w is aw, where a is the with root of unity defined by em m. Taking
then the various branches as given by w, aw, a?w, ...... , am~lw, we have the law of inter

change for description of a small curve round any one-branch point as given by this
succession in cyclical order. The law of succession for a circuit enclosing more than
one of the branch-points is derivable by means of Corollary V, § 90.

To find the relation of z = o> to w, we take zz' = l and consider the new function W in
the vicinity of the ^'-origin. We have

W ={A (1 -VH1 -a/) ...... (l-an<)}^'~»».

If the variable z1 describe a very small circle round the origin in the negative sense, then

27TZ —

z' is multiplied by e~2™ and so W acquires a factor e ™, that is, W is changed unless
this acquired factor is unity. It can be unity only when n/m is an integer ; and therefore,
except when n/m is an integer, 0=00 is a branch-point of the function. The law of
succession is the same as that for negative description of the z'-circle, viz., w, anw,
a2nw, ...... ; the m values form a single cycle only if n be prime to m, and a set of cycles

if n be not prime to m.

Thus 0=00 is a branch -point for w = (k?-gg-g^~^ ; it is not a branch-point for
w = {(\ -22) (1 — &2z2)}~*; and z = b is a branch-point for the function defined by

(z — b) w2 = z — a,
but z = b is riot a branch-point for the function defined by (z—b)2wz = z-a.

Again, if p denote a particular value of ft when z has a given value, and q similarly

denote a particular value of [— — : ) , then w=p+q is a six-valued function, the values

V+v
being

W6= -p + aq,

where a is a primitive cube root of unity. The branch-points are - 1, 0, 1, oo ; and the
orders of change for small circuits round one (and only one) of these points are as
follows :

For a small circuit round

-1

0

1

00

Wj changes to

•ft

W-i

W3

W2

W2 „

we

W1

W4

w,

^3

to,

W4

W5

W4

W4 „

M>2

w,

W6

W3

Ws n

W3

W6

w,

W6

U>6

W74

W5

Wo

W5

93.] FUNCTIONS 161

Combinations can at once be effected ; thus, for a positive circuit enclosing both 1 and QO
but* not — 1 or 0, the succession is

iolt w4, w6, w2, w3, WG
in cyclical order.

94. It has already been remarked that algebraic irrationals are a special
class of functions denned by algebraical equations. Functions thus generally
denned by equations, which are algebraical so far as concerns the dependent
variable but need not be so in reference to the independent variable, are
often called algebraical. The term, in one sense, cannot be strictly applied
to the roots of an equation of every degree, seeing that the solution
of equations of the fifth and higher degrees can be effected only by
transcendental functions; but what is implied is that a finite number of
determinations of the dependent variable is given by the equation -f*.

The equation is algebraical in relation to the dependent variable w, that
is, it will be taken to be of finite degree n in w. The coefficients of the
different powers will be supposed to be rational uniform functions of z : were
they irrational in any given equation, the equation could be transformed
into another, the coefficients of which are rational uniform functions. And
the equation is supposed to be irreducible, that is, if the equation be taken
in the form

f(w, *) = 0,

the left-hand member f(w, z) cannot be resolved into factors of a form and
character as regards w and z similar to /itself.

The existence of equal roots of the equation for general values of z
requires that

fi \ j "df(w> z)
f(w,z) and ^~

shall have a common factor, which will be rational owing to the form of
f(w, z}. This form of factor is excluded by the irreducibility of the equation ;
so that /= 0, as an equation in w, has not equal roots for general values
of z. But though the two equations are not both satisfied in virtue of a
simpler equation, they are two equations determining values of w and #;
and their form is such that they will give equal values of w for special
values of z.

Since the equation is of degree n, it may be taken to be

w

where the functions F1} F2}... are rational and uniform. If all their singu-

* Such a circuit, if drawn on the Neumann's sphere, may be regarded as excluding - 1 and 0,
or taking account of the other portion of the surface of the sphere, it may be regarded as a
negative circuit including - 1 and 0, the cyclical interchange for which is easily proved to be
iCj, w4, w5, w.2, M?3, w6 as in the text.

t Such a function is called Men defini by Liouville.

F. 11

162 ALGEBRAICAL [94.

larities be accidental, they are raeromorphic algebraical functions of z (unless
z = oo is the only singularity, in which case they are holomorphic) ; and the
equation can then be replaced by one which is equivalent and has all its
coefficients holomorphic, the coefficient of wn being the least common multiple
of all the denominators of the meromorphic functions in the first form. This
form cannot however be deduced, if any of the singularities be essential.

The equation, as an equation in w, has n roots, all functions of z ; let
these be denoted by w1,w2,..., ivn, which are the n branches of the function w.
When the geometrical interpretation is associated with the analytical relation,
there are n points in the w-plane, say a1,..., an, which correspond with a point
in the ^-plane, say with c^ ; and in general these n points are distinct. As
z varies so as to move in its own plane from a, then each of the w-points
moves in their common plane ; and thus there are n w-paths corresponding
to a given z-path. These n curves may or may not meet one another.

If they do not, there are n distinct w-paths, leading from a1;..., an to
/3i,..., /3n, respectively corresponding to the single ^-path leading from a
to b.

If two or more of the w-paths do meet one another, and if the describing
w-poirits coincide at their point of intersection, then at such a point of
intersection in the w-plane, the associated branches w are equal ; and
therefore the point in the ^-plane is a point that gives equal values for w.
It is one of the roots of the equation obtained by the elimination of w
between

the analytical test as to whether the point is a branch-point will be
considered later. The march of the concurrent ^-branches from such a
point of intersection of two w-paths depends upon their relations in its
immediate vicinity.

When no such point lies on a ^-path from a to b, no two of the w-points
coincide during the description of their paths. By § 90, the 2-path can be
deformed (provided that, in the deformation, it does not cross a branch-point)
without causing any two of the w-points to coincide. Further, if z describe
a closed curve which includes none of the branch -points, then each of the
^-branches describes a closed curve and no two of the tracing points ever
coincide.

Note. The limitation for a branch-point, that the tracing w-points
coincide at the point of intersection of the w-curves, is of essential im
portance.

What is required to establish a point in the z-plane as a branch-point,
is not a mere geometrical intersection of a couple of completed w-paths but
the coincidence of the w-points as those paths are traced, together with inter-

94.] FUNCTIONS 1 63

change of the branches for a small circuit round the point. Thus let there be
such a geometrical intersection of two w-curves, without coincidence of the
tracing points. There are two points in the ^-plane corresponding to the
geometrical intersection ; one belongs to the intersection as a point of the
w-paih which first passed through it, and the other to the intersection as a
point of the w-path which was the second to pass through it. The two
branches of w for the respective values of z are undoubtedly equal ; but the
equality would not be for the same value of z. And unless the equality
of branches subsists for the same value of z, the point is not a branch
point.

A simple example will serve to illustrate these remarks. Let w be defined by the
equation

so that the branches w1 and w2 are given by

Ci0j_ = cz+z(z2 + c2)*, cw2 = cz-z(z*-\- c2)* ;
it is easy to prove that the equation resulting from the elimination of w between /=0 and

and that only the two points z= ±ic are branch-points.

The values of z which make wl equal to the value of wz for z = a (supposed not equal to
either 0, ci or — ci) are given by

cz + z (02 + c2)* = ca - a (a2 + c2)*,

which evidently has not 2 = a for a root. Rationalising the equation so far as concerns z
and removing the factor z -a, as it has just been seen not to furnish a root, we find that s
is determined by

z3 + z2a + za2 + a3 + 2ac2 - 2ac (a2 + c2) * = 0,

the three roots of which are distinct from a, the assumed point, and from ±ci, the branch
point. Each of these three values of z will make wv equal to the value of w2 for z=a : we
have geometrical intersection without coincidence of the tracing points.

95. When the characteristics of a function are required, the most im
portant class are its infinities: these must therefore now be investigated.
It is preferable to obtain the infinities of the function rather than the
singularities alone, in the vicinity of which each branch of the function
is uniform * : for the former will include these singularities as well as
those branch-points which, giving infinite values, lead to regular singularities
when the variables are transformed as in § 93. The theorem which deter
mines them is: —

The infinities of a function determined by an algebraical equation are the
singularities of the coefficients of the equation.
Let the equation be

wn + wn-i FI ^ + wn-, !»,(*) + ... + rf^ (^) + ^ (^) = Q,

* These singularities will, for the sake of brevity, be called regular.

11—2

164 INFINITIES [95.

and let w' be any branch of the function; then, if the equation which
determines the remaining branches be

wn-i + wn-2 Qi ^ + wn-3 £2 (Y) + . . . + WGn-z (Z) + Gn-i (z) = 0,

we have Fn (z) = - w'Gn-i (z),

Fn^ (z) = - w'Gn-z (z) + £„_! (z),

^71-2 (Z) = - w'Gn-s (z} + #n-2 (z),

Now suppose that a is an infinity of w' ; then, unless it be a zero of order
at least equal to that of Gn^ (z), a is an infinity of Fn (z). If, however, it be
a zero of Gn-i (z) of sufficient order, then from the second equation it is an
infinity of Fn_l(z) unless it is a zero of order at least equal to that of
6rn_2 (z) ; and so on. The infinity must be an infinity of some coefficient not
earlier than Fi (z) in the equation, or it must be a zero of all the functions
G which are later than Gf_! (z). If it be a zero of all the functions Gr, so
that we may not, without knowing the order, assert that it is of rank at
least equal to its order as an infinity of w', still from the last equation it
follows that a must be an infinity of Fl (z). Hence any infinity of w is an
infinity of at least one of the coefficients of the equation.

Conversely, from the same equations it follows that a singularity of one
of the coefficients is an infinity either of w' or of at least one of the co
efficients G. Similarly the last alternative leads to an inference that the
infinity is either an infinity of another branch w" or of the coefficients of the
(theoretical) equation which survives when the two branches have been
removed. Proceeding in this way, we ultimately find that the infinity either
is an infinity of one of the branches or is an infinity of the coefficient in the
last equation, that is, of the last of the branches. Hence any singularity
of a coefficient is an infinity of at least one of the branches of the function.

It thus appears that all the infinities of the function are included among,
and include, all the singularities of the coefficients ; but the order of the
infinity for a branch does not necessarily make that point a regular
singularity nor, if it be made a regular singularity, is the order necessarily
the same as for the coefficient.

96. The following method is effective for the determination of the order
of the infinity of the branch.

Let a be an accidental singularity of one or more of the F functions,
say of order ra; for the function Ft ; and assume that, in the vicinity of a,
we have

Ft (z) = (z- a)-™* [Ci + di (z-a) + e{ (z - a? +...].

96.]

OF ALGEBRAICAL FUNCTIONS

165

Then the equation which determines the first term of the expansion of w in
a series in the vicinity of a is

wn + d (z — a)~™i wn~l + c2(z — a)~m2 wn~2 + ...

-f cn_! (z - a)~m»-i w + cn (z - a)~m« = 0.

Mark in a plane, referred to two rectangular axes, points n, 0; n — 1,
— m^; n — 2, — m2 ; . . ., 0, — mn ; let these
be A0, A1} ..., An respectively. Any line
through Ai has its equation of the form

1 1 —I— nm • ~~ ~\ J o" — l w — T. t)\\
y T »*< — A, {J, (71, tftf

that is,

y — \x = — \ (n — i) — mi.

If then w = (z — a)~xf(z}, where f(z) is
finite when z = a, the intercept of the fore
going line on the negative side of the axis of y is equal to the order of the
infinity in the term

wn-iFi(z).

This being so, we take a line through An coinciding in direction with the
negative part of the axis of y and we turn it about An in a trigonometrically
positive direction until it first meets one of the other points, say An_r ; then
we turn it about An_r until it meets one of the other points, say An_s; and
so on until it passes through A0. There will thus be a line from An to
A0, generally consisting of a number of parts ; and none of the points A
will be outside it.

The perpendicular from the origin on the line through An_r and An_g is
evidently greater than the perpendicular on any parallel line through a
point A, that is, on any line through a point A with the same value
of X; and, as this perpendicular is

it follows the order of the infinite terms in the equation, when the particular
substitution is made for w, is greater for terms corresponding to points lying
on the line than it is for any other terms.

If /(*) = 0 wnen z = a, then the terms of lowest order after the substitu
tion of (z — a)~Kf(z) for w are

as many terms occurring in the bracket as there are points A on the line
joining An_r to An_s. Since the equation determining w must be satisfied,
terms of all orders must disappear, and therefore

an equation determining s-r values of 6, that is, the first terms in the
expansions of s — r branches w.

166 INFINITIES [96.

Similarly for each part of the line : for the first part, there are r branches
with an associated value of X ; for the second, s — r branches with another
associated value ; for the third, t — s branches with a third associated value ;
and so on.

The order of the infinity for the branches is measured by the tangent
of the angle which the corresponding part of the broken line makes with the
axis of a; ; thus for the line joining An^. to An_s the order of the infinity for
the s - r branches is

where mn_r and mn_s are the orders of the accidental singularities of Fn_r (z)
and Fn_s (z).

If any part of the broken line should have its inclination to the axis of
x greater than \ir so that the tangent is negative and equal to - //,, then the
form of the corresponding set of branches w is (z — a,y g {z} for all of them,
that is, the point is not an infinity for those branches. But when the
inclination of a part of the line to the axis is < \TT, so that the tangent is
positive and equal to X, then the form of the corresponding set of branches
w is (z — a)~Kf(z) for all of them, that is, the point is an infinity of order X
for those branches.

In passing from An to A0 there may be parts of the broken line which
have the tangential coordinate negative, implying therefore that a is not an
infinity of the corresponding set or sets of branches w. But as the revolving
line has to change its direction from Any' to some direction through A0,
there must evidently be some part or parts of the broken line which have
their tangential coordinate positive, implying therefore that a is an infinity
of the corresponding set or sets of branches.

Moreover, the point a is, by hypothesis, an accidental singularity of at
least one of the coefficients and it has been supposed to be an essential
singularity of none of them; hence the points A0, A1} ..., An are all in the
finite part of the plane. And as no two of their abscissa are equal, no line
joining two of them can be parallel to the axis of y, that is, the inclination
of the broken line is never \ir and therefore the tangential coordinate is
finite, that is, the order of the infinity for the branches is finite for any
accidental singularity of the coefficients.

If the singularity at a be essential for some of the coefficients, the
corresponding result can be inferred by passing to the limit which is
obtained by making the corresponding value or values of m infinite. In
that case the corresponding points A move to infinity and then parts of the
broken line pass through A0 (which is always on the axis of x) parallel to
the axis of y, that is, the tangential coordinate is infinite and the order of

96.] OF ALGEBRAICAL FUNCTIONS 167

the infinity at a for the corresponding branches is also infinite. The point is
then an essential singularity (and it may be also a branch-point).

It has been assumed implicitly that the singularity is at a finite point in
the 2-plane ; if, however, it be at oo , we can, by using the transformation
zz' — 1 and discussing as above the function in vicinity of the origin, obtain
the relation of the singularity to the various branches. We thus have the
further proposition :

The order of ike infinity of a branch of an algebraical function at a
singularity of a coefficient of the equation, which determines the function, is
finite or infinite according as the singularity is accidental or essential.

If the coefficients FI of the equation be holomorphic functions, then
z = oo is their only singularity and it is consequently the only infinity for
branches of the function. If some of or all the coefficients Ff be mero-
morphic functions, the singularities of the coefficients are the zeros of
the denominators and, possibly, £=oo; and, if the functions be algebraical,
all such singularities are accidental. In that case, the equation can be
modified to

h0 (z) wn + h^ (z} wn~l + A2 (z) wn~2 + . . . = 0,

where h0(z) is the least common multiple of all the denominators of the
functions Ft. The preceding results therefore lead to the more limited
theorem :

When a function w is determined by an algebraical equation the coefficients
of which are holomorphic functions of z, then each of the zeros of the coefficient
of the highest power of w is an infinity of some of (and it may be of all) the
branches of the function w, each such infinity being of finite order. The point
z= oo may also be an infinity of the function w ; the order of that infinity is
finite or infinite according as z = oo is an accidental or an essential singularity
of any of the coefficients.

It will be noticed that no precise determination of the forms of the
branches w at an infinity has been made. The determination has, however,
only been deferred : the infinities of the branches for a singularity of the
coefficients are usually associated with a branch-point of the function and
therefore the relations of the branches at such a point will be of a general
character independent of the fact that the point is an infinity.

If, however, in any case a singularity of a coefficient should prove to be,
not a branch-point of w but only a regular singularity, then in the vicinity of
that point the branch of w is a uniform function. A necessary (but not suffi
cient) condition for uniformity is that (mn_r — mn_s) -7- (s — r) be an integer.

Note. The preceding method can be applied to determine the leading
terms of the branches in the vicinity of a point a which is an ordinary point
for each of the coefficients F.

168 BRANCH-POINTS [97.

97. There remains therefore the consideration of the branch-points of a
function determined by an algebraical equation.

The characteristic property of a branch-point is the equality of branches
of the function for the associated value of the variable, coupled with the
interchange of some of (or all) the equal branches after description by the
variable of a small contour enclosing the point.

So far as concerns the first part, the general indication of the form of the
values has already (§ 93) been given. The points, for which values of w
determined as a function of z by the equation

f(w, z) =J0

are equal, are determined by the solution of this equation treated simul
taneously with

df(w, z) = Q.
dw

and when a point z is thus determined the corresponding values of w, which
are equal there, are obtained by substituting that value of z and taking M,

the greatest common measure of / and -J- . The factors of M then lead to

the value or the values of w at the point ; the index m of a linear factor
gives at the point the multiplicity of the value which it determines, and
shews that m + 1 values of w have a common value there, though they are
distinct at infinitesimal distances from the point. If m = 1 for any factor,
the corresponding value of w is an isolated value and determines a branch
that is uniform at the point.

Let z = a, w = a be a value of z and a value of w thus obtained ; and
suppose that m is the number of values of w that are equal to one another.
The point z = a is not a branch-point unless some interchange among the
in values of w is effected by a small circuit round a ; and it is therefore
necessary to investigate the values of the branches* in the vicinity of z — a.

Let w = a. + w', z = a + z' ; then we have

that is, on the supposition that f(w, z) has been freed from fractions,

/(a, a) + SS^rXV = 0,

r, s

so that, since a is a value of w corresponding to the value a of z, we have
w' and / connected by the relation

* The following investigations are founded on the researches of Puiseux on algebraic
functions; they are contained in two memoirs, Liouville, lre Ser., t. xv, (1850), pp. 365 — 480, ib.,
t. xvi, (1851), pp. 228—240. See also the chapters on algebraic functions, pp. 19 — 76, in the
second edition of Briot and Bouquet's Theorie des fonctions elliptiques.

97.] OF ALGEBRAICAL FUNCTIONS 169

When / is 0, the zero value of w' must occur m times, since a is a root
m times repeated; hence there are terms in the foregoing equation inde

pendent of z, and the term of lowest index among them is w'm. Also when
w ' = 0, z' — 0 is a possible root ; hence there must be a term or terms
independent of w' in the equation.

First, suppose that the lowest power of z among the terms independent
of w' is the first. The equation has the form

Az' + higher powers of z'
+ Biu' + higher powers of w'
+ terms involving z' and w' = 0,

O-/* / \

where A is the value of - ' — - for w = a, z = a. Let z'=%m, w' = v%: the

02

last form changes to

(A + Bvm) £m + terms with £m+1 as a factor = 0 ;
and therefore A + Bvm + terms involving £= 0.

Hence in the immediate vicinity of z = a, that is, of £ =0, we have

A + Bv™ = 0.

Neither A nor B is zero, so that all the m values of v are finite. Let them
be vl}..., vm, so arranged that their arguments increase by 2-Tr/Tn through
the succession. The corresponding values of w' are

for i = l, ..., m. Now a ^-circuit round a, that is, a /-circuit round its
origin, increases the argument of z' by 2?r ; hence after such a circuit we

1_ 27Tt !_

have the new value of w{ as ViZ/m em, that is, it is vi+1z'm which is the value
of w'i+l. Hence the set of values w\, «/.,,..., w'm form a complete set of
interchangeable values in their cyclical succession ; all the m values, which
are equal at a, form a single cycle and the point is a branch-point.

Next, suppose that the lowest power of z among the terms independent
of w is z' , where I > 1. The equation now has the form
0 = Az' + higher powers of z'
+ Bw' + higher powers of w'

Arsz'V

r=l s=l

where in the last summation r and s are not zero and in every term either
(i), r is equal to or greater than I or (ii), s is equal to or greater than m
or (iii), both (i) and (ii) are satisfied. As only terms of the lowest orders

170

BRANCH-POINTS

[97.

need be retained for the present purpose, which is the derivation of the first
term of w' in its expansion in powers of z', we may use the foregoing equation
in the form

, l-lm-l

A/ + 2 2 A,

r=l s=l

,r ,s , -p. ,m _

jf w + Bw = 0.

To obtain this first term we proceed in a manner similar to that in § 96 *.
Points A0,..., Am are taken in a plane
referred to rectangular axes having as co
ordinates 0, £;...; s, r;...; m, 0 respectively.
A line is taken through Am and is made to
turn round Am from the position AmO until
it first meets one of the other points ; then
round the last point which lies in this
direction, say round Aj, until it first meets
another ; and so on.

Any line through At (the point si} rt) is
of the form

y - Ti = - \ (x - s^.

The intercept on the axis of /-indices is \Si + Ti, that is, the order of the
term involving Ars for a substitution w' oc / . The perpendicular from the
origin for a line through AI and Aj is less than for any parallel line through
other points with the same inclination ; and, as this perpendicular is

Fig. 21.

it follows that, for the particular substitution w' oc z' , the terms corresponding
to the points lying on the line with coordinate X are the terms of lowest
order and consequently they are the terms which give the initial terms for
the associated set of quantities w'.

Evidently, from the indices retained in the equation, the quantities X
for the various pieces of the broken line from Am to A0 are positive and
finite.

Consider the first piece, from Am to Aj say ; then taking the value of X for

that piece as fa, so that we write v^z'*1 as the first term of w', we have as the
set of terms involving the lowest indices

J? /"* i ^ ^ A fl* fi I A fl*i , ^J

Sj being the smallest value of s retained ; and then

so that

/*! =

m — s

* Reference in this connection may be made to Chrystal's Algebra, ch. xxx., with great
advantage, as well as the authorities quoted on p. 168, note.

GROUPING OF BRANCHES 171

Let p/q be the equivalent value of ^ as the fraction in its lowest terms ; and

p

write / = (?. Then w' = vlz'i = vtf ; all the terms except the above group

are of order > mp and therefore the equation leads after division by %mPtfi to

Bv^-'i + ^Aravf-*i + Arfj = 0,

an equation which determines m — Sj values for vl, and therefore the initial
terms of m — Sj of the w-branches.

Consider now the second piece, from Aj to At say ; then taking the value

of A, for that piece as fa, so that we write v.2z'^ as the first term of w', we
have as the set of terms involving the lowest indices for this value of /*2

A fri /s.- xr"O A iv Is fl"i tsi

Arfz' Jw ' + E&A.rjt w' + Ar.sz' *w \
where S{ is the smallest value of s retained. Then

Sjfr + Tj = tyig + r

Proceeding exactly as before, we find

as the equation determining Sj-Si values for v2 and therefore the initial
terms of Sj — st of the w-branches.

And so on, until all the pieces of the line are used ; the initial terms of
all the w-branches are thus far determined in groups connected with the
various pieces of the line A^Ai^.A,. By means of these initial terms,
the m-branches can be arranged for their interchanges, by the description of
a small circuit round the branch-point, according to the following theorem :—

Each group can be resolved into systems, the members of each of which are
cyclically interchangeable.

It will be sufficient to prove this theorem for a single group, say the
group determined by the first piece of broken line: the argument is
general.

Since - is the equivalent of — ^— and of T} . and since s, < s, we have
V m — s m — Sj

m-s = kq, m-Sj^kjq, kj>k;
and then the equation which determines ^ is

Sv&v + 2^r,,Vl <*,-*> 1 4 ArjSj = 0,

that is, an equation of degree k} in vj as its variable. Let U be any root of
it ; then the corresponding values of vl are the values of U*. Suppose these
q values to be arranged so that the arguments increase by 27r^, which is

possible, because p is prime to q. Then the q values of w' being the values
of v^Vi are

P. P P

172 GROUPING OF BRANCHES [97.

where vla is that value of Ifi which has — — for its argument. A circuit

round the /-origin evidently increases the argument of any one of these
w'-values by Zrrp/q, that is, it changes it into the value next in the succession;
and so the set of q values is a system the members of which are cyclically
interchangeable.

This holds for each value of U derived from the above equation ; so that
the whole set of m — Sj branches are resolved into kj systems, each containing
q members with the assigned properties.

It is assumed that the above equation of order kj in vj has its roots unequal.
If, however, it should have equal roots, it must be discussed ab initio by a
method similar to that for the general equation; as the order kj (being a
factor of m — Sj) is less than m, the discussion will be shorter and simpler,
and will ultimately depend on equations with unequal roots as in the case
above supposed.

It may happen that some of the quantities /j, are integers, so that the
corresponding integers q are unity : a number of the branches would then be
uniform at the point.

It thus appears that z = a is a branch-point and that, under the present
circumstances, the m branches of the function can be arranged in systems,
the members of each one of which are cyclically interchangeable.

Lastly, it has been tacitly assumed in what precedes that the common
value of w for the branch-point is finite. If it be infinite, this infinite value
can, by § 95, arise only out of singularities of the coefficients of the equation :
and there is therefore a reversion to the discussion of §§ 95, 96. The dis
tribution of the various branches into cyclical systems can be carried out
exactly as above.

Another method of proceeding for these infinities would be to take
ww' = \, z= c + z' ; but this method has no substantial advantage over the
earlier one and, indeed, it is easy to see that there is no substantial
difference between them.

Ex. 1. As an example, consider the function determined by the equation

The equation determining the values of z which give equal roots for w is

82 (2 -1)2 = 4(3 -I)3
so that the values are z=l (repeated) and z= — 1.

When z=l, then w=0, occurring thrice; and, if 2 = 1+2' then

8W/3W,

that is, w'^^z13.

The three values are branches of one system in cyclical order for a circuit round z=\.

97.] EXAMPLES 173

When z = — 1, the equation for w is

that is, (w

so that w=\ or w= — £, occurring twice.

For the former of these we easily find that, for s= — l-\-z', the value of w is
l-hfs'-f ...... , an isolated branch as is to be expected, for the value 1 is not

repeated.

For the latter we take w—— \ + w' and find

so that the two branches are

and they are cyclically interchangeable for a small circuit round z= - 1.

These are the finite values of w at branch-points. For the infinities of w, which may
arise in connection with the singularities of the coefficients, we take the zeros of the
coefficient of the highest power of w in the integral equation, viz., 2 = 0, which is thus the
only infinity of w. To find its order we take w=z~nf (z)—yz~n + ...... , where y is a

constant and f (z) is finite for 2 = 0; and then we have

8zl~3n

J. "~

Thus l-3n=-n,

provided both of them be negative; the equality gives n = \ and satisfies the condition.
And 8y3= - 3y. Of these values one is zero, and gives a branch of the function without
an infinity; the other two are ±^V-f and they give the initial term of the two
branches of w, which have an infinity of order -^ at the origin and are cyclically
interchangeable for a small circuit round it. The three values of w for infinitesimal
values of z are

3 . _i 1 7 /3 . l 4 275
" - 81 '-1944

3 • -

M + — /?&*— 1*4. 215 /3-f_jL2_

6 18 V 8 81 1944 V 8 729 z

_ _i A As
w3--g + gj2+— 2 +

The first two of these form the system for the branch-point at the origin, which is neither
an infinity nor a critical point for the third branch of the function.

Ex. 2. Obtain the branch-points of the functions which are defined by the following
equations, and determine the cyclical systems at the branch-points :
(i) w*
(ii) w
(iii) w
(iv) iff

44
(v) vfi - (1 - a2) 104 _ _ Z2 (! _ 22)4 = 0- (Briot and Bouquet.)

Also discuss the branches, in the vicinity of 2 = 0 and of 2=00, of the functions defined
by the following equations :

(vi) aw7 + bu£z + cutz* + dwW + ewz1 +fz9 + gv£ + hw*£ + kzw = 0 ;
(vii) wmzn+wn+zm = Q.

174 SIMPLE BRANCH-POINTS [98.

98. There is one case of considerable importance which, though limited
in character, is made the basis of Clebsch and Gordan's investigations* in the
theory of Abelian functions — the results being, of course, restricted by the
initial limitations. It is assumed that all the branch-points are simple, that
is, are such that only one pair of branches of w are interchanged by a circuit
of the variable round the point ; and it is assumed that the equation /= 0 is
algebraical not merely in w but also in z. The equation f = 0 can then be
regarded as the generalised form of the equation of a curve of the nth order,
the generalisation consisting in replacing the usual coordinates by complex
variables; and it is further assumed, in order to simplify the analysis, that all
the multiple points on the curve are (real or imaginary) double-points. But,
even with the limitations, the results are of great value : and it is therefore
desirable to establish the results that belong to the present section of the
subject.

We assume, therefore, that the branch-points are such that only one
pair of branches of w are interchanged by a small closed circuit round any
one of the points. The branch-points are among the values of z determined
by the equations

z) A
>

When /=0 has the most general form consistent with the assigned
limitations, f (w, z) is of the ?ith degree in z ; the values of z are determined
by the eliminant of the two equations which is of degree n(n — 1), and there
are, therefore, n(n — Y) values of z which must be examined.

First, suppose that J \, ' — ' does not vanish for a value of z, thus

oz

obtained, and the corresponding value of w : then we have the first case
in the preceding investigation. And, on the hypothesis adopted in the
present instance, m = 2 ; so that each such point z is a branch-point.

Next, suppose that — ^ - vanishes for some of the n(n — 1) values of z ;

the value of m is still 2, owing to the hypothesis. The case will now be still

d'2f (w z}
further limited by assuming that ^ .2 does not vanish for the value of z

and the corresponding value of w ; and thus in the vicinity of z = a, w = a we
have an equation

0 = Az- + 2Bz'w' + Cw'2 -f terms of the third degree + ...... ,

where A, B, C are the values of ^ , =-£- , «~ f°r z — a> w=a.

oz1 dzdw ow2

If B2 AC, this equation leads to the solution

C'w + Bz oc uniform function of z.

* Clebsch und Gordan, Theorie der AbeVschen Functionen, (Leipzig, Teubner, 1866).

98.] SIMPLE BRANCH-POINTS 175

The point z = a, w = a is not a branch-point ; the values of w, equal at the
point, are functionally distinct. Moreover, such a point z occurs doubly in
the eliminant; so that, if there be B such points, they account for 28 in
the eliminant of degree n (n — 1) ; and therefore, on their score, the number
n (n — 1) must be diminished by '28. The case is, reverting to the genera
lisation of the geometry, that of a double point where the tangents are
not coincident.

If, however, B2 = AC, the equation leads to the solution

Cw' + Bz' = Lz'^ + Mz'* + Nz'* +

The point z = a, w = a is a point where the two values of z interchange.
Now such a point z occurs triply in the eliminant ; so that, if there be K
such points, they account for SK of the degree of the equation. Each of
them provides only one branch-point, and the aggregate therefore provides K
branch-points ; hence, in counting the branch-points of this type as derived
through the eliminant, its degree must be diminished by 2/c. The case is,
reverting to the generalisation of the geometry, that of a double point (real
or imaginary) where the tangents are coincident.

It is assumed that all the n(n— 1) points z are accounted for under
the three classes considered. Hence the number of branch-points of the
equation is

£l = n (n - 1) - 28 - 2«,

where n is the degree of the equation, B is the number of double points
(in the generalised geometrical sense) at which tangents to the curve do not
coincide, and K is the number of double points at which tangents to the
curve do coincide.

And at each of these branch-points, II in number, two branches of the
function are equal and, for a small circuit round it, interchange.

99. The following theorem is a combined converse of many of the
theorems which have been proved :

A function w, which has n (and only ?/) values for each value of z, and
which has a finite number of infinities and of branch-points in any part of the
plane, is a root of an equation in w of degree n, the coefficients of which are
uniform functions of z in that part of the plane.

We shall first prove that every integral symmetric function of the n
values is a uniform function in the part of the plane under consideration.

n

Let Sk denote 2, w£, where k is a positive integer. At an ordinary point

i-\

of the plane, Sk is evidently a one-valued function and that value is finite ;
Sk is continuous ; and therefore the function Sk is uniform in the immediate
vicinity of an ordinary point of the plane.

176 FUNCTIONS POSSESSING [99.

For a point a, which is a branch-point of the function w, we know that
the branches can be arranged in cyclical systems. Let w1,..., w^ be such a
system. Then these branches interchange in cyclical order for a description
of a small circuit round a ; and, if z — a = Z*, it is known (§ 93) that, in the
vicinity of Z = 0, a branch w is a, uniform function of Z, say

Therefore wk = Gk ) + Pk (Z)

\£il

in the vicinity of Z = 0 ; say

w* = Ak + 2Bk>mZ-™ + 2 Ck>mZ™.

m=l m=l

Now the other branches of the function which are equal at a are derivable
from any one of them by taking the successive values which that one
acquires as the variable describes successive circuits round a. A circuit
of w round a changes the argument of z — a, by 27r. and therefore gives Z
reproduced but multiplied by a factor which is a primitive /xth root of unity,
say by a factor a ; a second circuit will reproduce Z with a factor a2 ; and so
on. Hence

wf = Ak+2 Bk>m a— Z-™ +ZCk>m a- #»

wrk = Ak+2 Bk>m a-™ Z~m + 2 Ck,m a™ Zm,

m=l »»=!

and therefore
I*

wrk = pAk + 2 Bkm - + ar + cr + . . . + cr't

r=l m = 1

+ 2 flto* Zm (1 + «m + a2"* + • • • + a""*-™).

OT = 1

Now, since a is a primitive /*th root of unity,

1 +as + «2S+ ... + as('x-1)

is zero for all integral values of s which are not integral multiples of p,, and it
is yu, for those values of s which are integral values of jj, ; hence

- £

B'k> i(z - a)"1 + B'k^(z - a)~2 + B'kt3 (z - a)

.
Hence the point z = a may be a singularity of 2 wrk but it is not a branch-

r=l

99.] A FINITE NUMBER OF BRANCHES 177

point of the function ; and therefore in the immediate vicinity of z — a the

*i
quantity X wrk is a uniform function.

r=l

The point a is an essential singularity of this uniform function, if the
order of the infinity of w at a be infinite : it is an accidental singularity, if
that order be a finite integer.

This result is evidently valid for all the cyclical systems at a, as well as
for the individual branches which may happen to be one-valued at a. Hence

(U.

Sk, being the sum of sums of the form 2) wrk each of which is a uniform

r=l

function of z in the vicinity of a, is itself a uniform function of z in that
vicinity. Also a is an essential singularity of Sk, if the order of the infinity at
z = a for any one of the branches of w be infinite ; and it is an accidental
singularity of Sk> if the order of the infinity at z = a for all the branches of w
be finite. Lastly, it is an ordinary point of Sk, if there be no branch of w for
which it is an infinity. Similarly for each of the branch-points.

Again, let c be a regular singularity of any one (or more) of the branches
of w ; then c is a regular singularity of every power of each of those branches,
the singularities being simultaneously accidental or simultaneously essential.
Hence c is a singularity of 8k : and therefore in the vicinity of c, $& is a
uniform function, having c for an accidental singularity if it be so for each of
the branches w affected by it, and having c for an essential singularity if it be
so for any one of the branches w.

It thus appears that in the part of the plane under consideration the
function 8k is one-valued ; and it is continuous and finite, except at certain
isolated points each of which is a singularity. It is therefore a uniform
function in that part of the plane ; and the singularity of the function at any
point is essential, if the order of the infinity for any one of the branches w at
that point be infinite, but it is accidental, if the order of the infinity for all the
branches w there be finite. And the number of these singularities is finite,
being not greater than the combined number of the infinities of the function
w, whether regular singularities or branch-points.

Since the sums of the kth powers for all positive values of the integer k
are uniform functions and since any integral symmetric function of the n
values is a rational integral algebraical function of the sums of the powers, it
follows that any integral symmetric function of the n values is a uniform
function of z in the part of the plane under consideration ; and every infinity
of a branch w leads to a singularity of the symmetric function, which is
essential or accidental according as the orders of infinity of the various
branches are not all finite or are all finite.

F. 12

178 FUNCTIONS POSSESSING [99.

Since w has n (and only n) values wlt... ,wn for each value of z, the
equation which determines w is

(W - Wj) (W-W2) ... (W- Wn) = 0.

The coefficients of the various powers of w are symmetric functions of the
branches wl , . . . , wn; and therefore they are uniform functions of z in the
part of the plane under consideration. They possess a finite number of
singularities, which are accidental or essential according to the character of
the infinities of the branches at the same points.

COROLLARY. If all the infinities of the branches in the finite part of the
whole plane be of finite order, then the finite singularities of all the coefficients
of the powers of w in the equation satisfied by w are all accidental ; and the
coefficients themselves then take the form of a quotient of an integral uniform
function (which may be either transcendental or algebraical, in the sense of
§ 47) by another function of a similar character.

If z = oc be an essential singularity for at least one of the coefficients,
through being an infinity of unlimited order for a branch of w, then one
or both of the functions in the quotient-form of one at least of the coefficients
must be transcendental.

If z = oo be an accidental singularity or an ordinary point for all the
coefficients, through being either an infinity of finite order or an ordinary
point for the branches of w, then all the functions which occur in all the
coefficients are rational, algebraical expressions. When the equation is
multiplied throughout by the least common multiple of the denominators
of the coefficients, it takes the form

wnh0 (z) + wn~* A, (z) + . . . + w hn_, (z} + hn (z) = 0,

where the functions h0(z), h^(z\ ..., hn(z) are rational, integral, algebraical
functions of z, in the sense of § 47.

A knowledge of the number of infinities of w gives an upper limit of the
degree of the equation in z in the last form. Thus, let at be a regular
singularity of the function ; and let Oi, fa, ji, ... be the orders of the infinities
of the branches at at- ; then

w^w-i ... wn(z — at )A',
where \ denotes Oi + fti + % + ..., is finite (but not zero) for z = at.

Let Ci be a branch-point, which is an infinity; and let p, branches w form a

ft
system for ct-, such that w(z — Cf)^ is finite (but not zero) at the point; then

w:w2 ... Wp (z — Q) '
is finite (but not zero) at the point, and therefore also

99.] A FINITE NUMBER OF BRANCHES 179

is finite, where Qit (/>;, ^i, ... are numbers belonging to the various systems;
or, if ei denote 0; + $f + tyi + . . . , then

Wl...Wn(z- Ci)6i

is finite for z = C;. Similarly for other symmetric functions of w.

Hence, if «j, a2, ... be the regular singularities with numbers X1; X2, ...
defined as above, and if c^ c2, ... be the branch -points, that are also infinities,
with numbers e1; e2, ... defined as above, then the product

(w-Wj) ...... (w-wn) n 0-af)A< n 0-Ci)e<

i=l 1=1

is finite at all the points ai and at all the points c;. The points a and the
points c are the only points in the finite part of the plane that can make the
product infinite : hence it is finite everywhere in the finite part of the plane,
and it is therefore an integral function of z.

Lastly, let p be the number for z = oo corresponding to \i for af or to e^
for C;, so that for the coefficient of any power of w in (w — w^) ...(w — wn) the
greatest difference in degree between the numerator and the denominator is
p in favour of the excess of the former.

Then the preceding product is of order

which is therefore the order of the equation in z when it is expressed in a
holomorphic form.

12—2

CHAPTER IX.

PERIODS OF DEFINITE INTEGRALS, AND PERIODIC FUNCTIONS IN GENERAL.

100. INSTANCES have already occurred in which the value of a function
of z is not dependent solely upon the value of z but depends also on the
course of variation by which z obtains that value ; for example, integrals of
uniform functions, and multiform functions. And it may be expected that,
a fortiori, the value of an integral connected with a multiform function will
depend upon the course of variation of the variable z. Now as integrals
which arise in this way through multiform functions and, generally, integrals
connected with differential equations are a fruitful source of new functions,
it is desirable that the effects on the value of an integral caused by variations
of a £-path be assigned so that, within the limits of algebraic possibility, the
expression of the integral may be made completely determinate.

There are two methods which, more easily than others, secure this result ;
one of them is substantially due to Cauchy, the other to Riemann.

The consideration of Riemann's method, both for multiform functions and
for integrals of such functions, will be undertaken later, in Chapters XV.,
XVI. Cauchy's method has already been used in preceding sections relating
to uniform functions, and it can be extended to multiform functions. Its
characteristic feature is the isolation of critical points, whether regular
singularities or branch-points, by means of small curves each containing one
and only one critical point.

Over the rest of the plane the variable z ranges freely and, under certain
conditions, any path of variation of z from one point to another can, as will
be proved immediately, be deformed without causing any change in the
value of the integral, provided that the path does not meet any of the small
curves in the course of the deformation. Further, from a knowledge of the
relation of any point thus isolated to the function, it is possible to calculate
the change caused by a deformation of the £-path over such a point; and
thus, for defined deformations, the value of the integral can be assigned
precisely.

100.] INTEGRAL OF A BRANCH 181

The properties proved in Chapter II. are useful in the consideration of
the integrals of uniform functions ; it is now necessary to establish the
propositions which give the effects of deformation of path on the integrals
of multiform function. The most important of these propositions is the
following : —

fb
If w be a multiform function, the value of I wdz, taken between two

J a

ordinary points, is unaltered for a deformation of the path, provided that the
initial branch of w be the same and that no branch-point or infinity be crossed
in the deformation.

Consider two paths acb, adb, (fig. 16, p. 152), satisfying the conditions
specified in the proposition. Then in the area between them the branch w
has no infinity and no point of discontinuity ; and there is no branch-point
in that area. Hence, by § 90, Corollary VI., the branch w is a uniform
monogenic function for that area; it is continuous and finite everywhere
within it and, by the same Corollary, we may treat w as a uniform, mono
genic, finite and continuous function. Hence, by § 17, we have

rb ra

(c) I wdz + (d) wdz = 0,

J a J b

the first integral being taken along acb and the second along bda; and
therefore

rb ra rb

(c) wdz = — (d}\ wdz = (d) \ wdz,

Jo, J b J a

shewing that the values of the integral along the two paths are the same
under the specified conditions.

It is evident that, if some critical point be crossed in the deformation,
the branch w cannot be declared uniform and finite in the area and the
theorem of § 17 cannot then be applied.

COROLLARY I. The integral round a closed curve containing no critical
point is zero.

COROLLARY II. A curve round a branch-point, containing no other
critical point of the function, can be deformed into a loop
without altering the value of fwdz ; for the deformation
satisfies the condition of the proposition. Hence, when
the value of the integral for the loop is known, the
value of the integral is known for the curve.

COROLLARY III. From the proposition it is possible
to infer conditions, under which the integral fwdz round
the whole of any curve remains unchanged, when the whole
curve is deformed, without leaving an infinitesimal arc
common as in Corollary II.

182 INTEGRATION [100.

Let GDC', ABA' be the curves: join two consecutive points A A' to two
consecutive points (7(7. Then if the area CABA'C'DG
enclose no critical point of the function w, the value of
jwdz along CDC' is by the proposition the same as its
value along CABA'C'. The latter is made up of the
value along CA, the value along ABA', and the value
along AC', say

rA r rC'

I wdz + I wdz + w'dz, v.

Jc JB JA Ǥ.

where w' is the changed value of w consequent on the description of a simple
curve reducible to B (§ 90, Cor. II.).

Now since w is finite everywhere, the difference between the values of w
at A and at A' consequent on the description of ABA is finite : hence as
A A is infinitesimal the value of jwdz necessary to complete the value for
the whole curve B is infinitesimal and therefore the complete value can be

taken as the foregoing integral wdz. Similarly for the complete value

J B

along the curve D : and therefore the difference of the integrals round B and
round D is

rA rC'

I wdz + I w'dz,

J C J A'

rA

say (w — w') dz.

J c

In general this integral is not zero, so that the values of the integral
round B and round D are not equal to one another : and therefore the curve
D cannot be deformed into the curve B without affecting the value of jwdz
round the whole curve, even when the deformation does not cause the curve
to pass over a critical point of the function.

But in special cases it may vanish. The most important and, as a
matter of fact, the one of most frequent occurrence is that in which the
description of the curve B restores at A' the initial value of w at A. It
easily follows, by the use of § 90, Cor. II., that the description of D (as
suming that the area between B and D includes no critical point) restores
at C' the initial value of w at (7. In such a case, w = w' for corresponding
points on AC and A'C', and the integral, which expresses the difference, is
zero: the value of the integral for the curve B is then the same as that for D.
Hence we have the proposition : —

If a curve be such that the description of it by the independent variable
restores the initial value of a multiform function w, then the value of jwdz
taken round the curve is unaltered when the curve is deformed into any other
curve, provided that no branch-point or point of discontinuity of w is crossed
in the course of deformation.

100.] OF MULTIFORM FUNCTIONS 183

This is the generalisation of the proposition of § 19 which has thus far
been used only for uniform functions.

Note. Two particular cases, which are very simple, may be mentioned
here : special examples will be given immediately.

The first is that in which the curve B, and therefore also D, encloses
no branch-point or infinity; the initial value of w is restored after a
description of either curve, and it is easy to see (by reducing B to a
point, as may be done) that the value of the integral is zero.

The second is that in which the curve encloses more than one branch
point, the enclosed branch-points being such that a circuit of all the loops,
into which (by Corollary V., § 90) the curve can be deformed, restores the
initial branch of w. This case is of especial importance when w is two-valued :
the curves then enclose an even number of branch-points.

101. It is important to know the value of the integral of a multiform
function round a small curve enclosing a branch-point.

Let c be a point at which TO branches of an algebraical function are equal
and interchange in a single cycle ; and let c, if an infinity, be of only finite
order, say k/m. Then in the vicinity of c, any of the branches w can be
expressed in the form

00 .«

w= 2 gs(z-c)m,

o — If

o — — K

where k is a finite integer.

The value of jwdz taken round a small curve enclosing c is the sum of
the integrals

the value of which, taken once round the curve and beginning at a point zly is

TO + S

where a is a primitive mth root of unity, provided TO + s is not zero. If then
s + m be positive, the value is zero in the limit when the curve is infini
tesimal : if TO + s be negative, the value is oo in the limit.

But, if m + s be zero, the value is Z7rigs.

Hence we have the proposition: If, in the vicinity of a branch-point c,
where m branches w are equal to one another and interchange cyclically, the
expression of one of the branches be

184 MULTIPLICITY OF VALUE [101.

then jwdz, taken once round a small curve enclosing c, is zero, if k<m; is
infinite, if k> m ; and is ^irig^ , if k = m.

It is easy to see that, if the integral be taken m times round the small
curve enclosing c, then the value of the integral is 2m7rigm when k is greater
than in, so that the integral vanishes unless there be a term involving (z — c)"1
in the expansion of a branch w in the vicinity of the point. The reason that
the integral, which can furnish an infinite value for a single circuit, ceases to

_*

do so for m circuits, is that the quantity (^ — c) m, which becomes indefi
nitely great in the limit, is multiplied for a single circuit by a*— 1, for a
second circuit by a2A — aA, and so on, and for the mth circuit by awA — a(w~1)A,
the sum of all of which coefficients is zero.

Ex. The integral \{(z - a) (z - b) ... (z -f)}~* dz taken round an indefinitely small curve
enclosing a is zero, provided no one of the quantities b, ... ,/ is equal to a.

102. Some illustrations have already been given in Chapter II., but
they relate solely to definite, not to indefinite, integrals of uniform
functions. The whole theory will not be considered at this stage ; we shall
merely give some additional illustrations, which will shew how the method
can be applied to indefinite integrals of uniform functions and to integrals
of multiform functions, and which will also form a simple and convenient
introduction to the theory of periodic functions of a single variable.

We shall first consider indefinite integrals of uniform functions.

f dz
Ex. 1. Consider the integral I — , and denote* it by/ (z}.

The function to be integrated is uniform, and it has an accidental singularity of the first
order at the origin, which is its only singularity. The value of \z~l dz taken positively
along a small curve round the origin, say round a circle with the origin as centre, is 2n-i •
but the value of the integral is zero when taken along any closed curve which does not
include the origin.

Taking z = l as the lower limit of the integral, and any point z as the upper limit, we
consider the possible paths from 1 to z. Any path from 1 to z can be deformed, without
crossing the origin, into a path which circumscribes the origin positively some number of
times, say m^, and negatively some number of times, say »i2, all in any order, and then leads
in a straight line from 1 to z. For this path the value of the integral is equal to

I — ,
J 1 z

that is, to 2mni+ I — ,

Ji z

where m is an integer, and in the last integral the variation of z is along a straight
line from 1 to z. Let the last integral be denoted by u ; then

* See Chrystal, ii, pp. 266 — 272, for the elementary properties of the function and its inverse,
when the variable is complex.

102.] OF INTEGRALS 185

and therefore, inverting the function and denoting/"1 by <j>, we have

Hence the general integral is a function of z with an infinite number of values ; and z is a
periodic function of the integral, the period being 2n-z.

Ex. 2. Consider the function / - - ^ > and again denote it by / (z).

The one- valued function to be integrated has two accidental singularities + i, each of
the first order. The value of the integral taken positively along a small curve round i is
TT, and along a small curve round — i is — n.

We take the origin 0 as the lower limit and any point z as the upper limit. Any path
from 0 to z can be deformed, without crossing either of the singularities and therefore
without changing the value of the integral, into

(i) any numbers of positive (ml5 w?2) an(* of negative (nz/, m2') circuits round i and
round -i, and

(ii) a straight line from 0 to z.
Then we have

- TJ-) +WIJJ ( - IT) + m.2' {_(-„•)}+ /* .

J o

, z
= nir+

where ?i is an integer and the integral on the right-hand side is taken along a straight line
from 0 to z.

Inverting the function and denoting/"1 by tp, we have

The integral, as before, is a function of z with an infinite number of values ; and z is a
periodic function of the integral, the period being TT.

103. Before passing to the integrals of multiform functions, it is con
venient to consider the method in which Hermite* discusses the multiplicity
in value of a definite integral of a uniform function.

Taking a simple case, let <£> (X) = \

J Q 1 + Z

and introduce a new variable t such that Z—zt\ then

zdt

When the path of t is assigned, the integral is definite, finite and unique in
value for all points of the plane except for those for which 1 + zt = 0 ; and,
according to the path of variation of t from 0 to 1, there will be a 0-curve
which is a curve of discontinuity for the subject of integration. Suppose the
path of t to be the straight line from 0 to 1 ; then the curve of discontinuity

* Crelle, t. xci, (1881), pp. 62—77; Cours a la Faculte des Sciences, 46me 6d. (1891), pp.
76—79, 154—164, and elsewhere.

186 HERMITE'S [103.

is the axis of x between — 1 and — oo . In this curve let any point - £ be
taken where £ > 1 ; and consider a point z1 — -^ + ie and a point z2 = — £ — ie,
respectively on the positive and the negative sides of the axis of x, both
being ultimately taken as infinitesimally near the point — £. Then

dt= (

Let e become infinitesimal ; then, when t is infinite, we have

tan

for e is positive ; and, when t is unity, we have

tan"1 ----- = — |TT,

for £ is > 1. Hence <£ (^) — <£ (^2)

The part of the axis of x from - 1 to - oo is therefore a line of discon
tinuity in value of <j> (z), such that there is a sudden change in passing from
one edge of it to the other. If the plane be cut along this line so that
it cannot be crossed by the variable which may not pass out of the plane,
then the integral is everywhere finite and uniform in the modified surface.
If the plane be not cut along the line, it is evident that a single passage
across the line from one edge to the other makes a difference of 2?ri in the
value, and consequently any number of passages across will give rise to the
multiplicity in value of the integral.

Such a line is called a section* by Hermite, after Riemann who, in a
different manner, introduces these lines of singularity into his method of
representing the variable on surfaces "f*.

When we take the general integral of a uniform function of Z and make
the substitution Z = zt, the integral that arises for consideration is of the form

We shall suppose that the path of variation of t is the axis of real quantities :
and the subject of integration will be taken to be a general function of t and
z, without special regard to its derivation from a uniform function of Z.

* Coupure; see Crelle, t. xci, p. 62. t See Chapter XV.

103.] SECTIONS 187

It is easy, after the special example, to see that ^ is a continuous function
of z in any space that does not include a ^-point which, for values of t included
within the range of integration, would satisfy the equation.

G (t, z) = 0.

But in the vicinity of a ^-point, say £, corresponding to the value t = 6 in
the range of integration, there will be discontinuity in the subject of
integration and also, as will now be proved, in the value of the integral.

Let Z be the point £ and draw the curve through Z corresponding to
t = real constant ; let Nt be a point on the positive side and N2
a point on the negative side of this curve positively described,
both points being on the normal at Z ; and let
supposed small. Then for N! we have

X-L = g — e sin y, yl = ^-\-e cos y ,

Fig. 24.
so that z1 = £+16' (cosy + isiny),

where ty is the inclination of the tangent to the axis of real quantities. But,
if da- be an arc of the curve at Z,

da , • • i \ d% • dt] d£

for variations along the tangent at Z, that is,

i

da- . . 3

-j-- (cos y + i sin y ) = — -

Thus, since -j- may be taken as finite on the supposition that Z is an
ordinary point of the curve, we have

where e = e' -y- , P = -

Similarly z.2 = £ + ie -^r.

Hence <1> (^) = I --i-i— *£ ^

w/ n_^J_w/ m _

1*.

188 HERMITE'S [103.

with a similar expression for <& (z2) ; and therefore

F(t, 0 j- [G (t, ®}^-G (t, §)

'

The subject of integration is infinitesimal, except in the immediate vicinity
of t = 6 ; and there

powers of small quantities other than those retained being negligible. Let
the limiting values of t, that need be retained, be denoted by d + v and
d — p', then, after reduction, we have

edt

F(e,

in the limit when e is made infinitesimal.

Hence a line of discontinuity of the subject of integration is a section
for the integral ; and the preceding expression is the magnitude, by
numerical multiples of which the values of the integral differ*.

Ex. 1. Consider the integral

dZ

/

zdt
h

We have S ^ *' =^ = ^g = ^.

so that TT is the period for the above integral.
Ex. 2. Shew that the sections for the integral

ta sin z ,
2 '

* The memoir and the Cmirs d' 'Analyse of Hermite should be consulted for further develop
ments; and, in reference to the integral treated above, Jordan, Cours d' Analyse, t. iii, pp.
610 — 614, may be consulted with advantage. See also, generally, for functions defined by
definite integrals, Goursat, Acta Math., t. ii, (1883), pp. 1—70, and ib., t. v, (1884), pp. 97—
120; and Pochhammer, Math. Arm., t. xxxv, (1890), pp. 470—494, 495—526. Goursat also
discusses double integrals.

103.] SECTIONS 189

where a is positive and less than 1, are the straight lines x = (2k + l) TT, where k assumes all
integral values ; and that the period of the integral at any section at a distance 77 from the
axis of real quantities is 2?r cosh (arj). (Hermite.)

Ex. 3. Shew that the integral

o

where the real parts of /3 and y — /3 are positive, has the part of the axis of real quantities
between 1 and +00 for a section.

Shew also that the integral

i

rht }— (z P~I (~i - vy~'3~1n— }~a d

J 0

where the real parts of /3 and 1 - a are positive, has the part of the axis of real quantities
between 0 and 1 for a section : but that, in order to render <£ (z) a uniform function of z,
it is necessary to prevent the variable from crossing, not merely the section, but also the
part of the axis of real quantities between 1 and + <x> . (Goursat.)

(The latter line is called a section of the second kind.)

Ex. 4. Discuss generally the effect of changing the path of t on a section of the

integral ; and, in particular, obtain the section for I — „ when, after the substitution

jo 1 + «

Z=zt, the path of t is made a semi-circle on the line joining 0 and 1 as diameter.

Note. It is manifestly impossible to discuss all the important bearings of theorems and
principles, which arise from time to time in our subject ; we can do no more than mention
the subject of those definite integrals involving complex variables, which first occur as
solutions of the better-known linear differential equations of the second order.

Thus for the definite integral connected with the hypergeometric series, memoirs by
Jacobi* and Goursat t should be consulted ; for the definite integral connected with
Bessel's functions, memoirs by HankelJ and Weber § should be consulted ; and Heine's
J/andbuch der Kugelfunctionen for the definite integrals connected with Legendre's
functions.

104. We shall now consider integrals of multiform functions.

Ex. 1. To find the integral of a multiform function round one loop ; and round a
number of loops.

Let the function be

i

w={(z-al}(z-a.z}...(z- an)}»» ,

where m may be a negative or positive integer, and the quantities a are unequal to one
another ; and let the loop be from the origin round the point ax. Then, if / be the value
of the integral with an assigned initial branch w, we have

/a, f CO

wdz-\- I wdz + I awdz,
0 J c J a.

where a is e m and the middle integral is taken round the circle at a^ of infinitesimal radius.

* Crelle, t. Ivi, (1859), pp. 149 — 165 ; the memoir was not published until after his death,
t Sur Vequation differentielle lineaire qui admet pour integrate la serie hypergrometrique,
(These, Gauthier-Villars, Paris, 1881).

I Math. Ann., t. i, (1869), pp. 467—501.

§ Math. Ann., t. xxxvii, (1890), pp. 404—416.

190

EXAMPLES

[104.

But, since the limit of (z-ajw when z = a1 is zero, the middle integral vanishes by § 101 ;
and therefore

/"«i
«, = (! -a) I web,

Jo

where the integral may, if convenient, be considered as taken along the straight line from
0 to al .

(2)

(3)

Fig. 25.

Next, consider a circuit for an integral of w which (fig. 25) encloses two branch-points,
say «! and «2, but no others ; the circuit in (1) can be deformed into that in (2) or into
that in (3) as well as into other forms. Hence the integral round all the three circuits
must be the same. Beginning with the same branch as in the first case, we have

(1

/«!
wdz,
o

as the integral after the first loop in (2). And the branch with which the second loop
begins is aw, so that the integral described as in the second loop is

/«2
awdz;
0

and therefore, for the circuit as in (2), the integral is

Cat [ay

1= (1 - a) I wdz + a (1 - a) / wdz.
Jo Jo

Proceeding similarly with the integral for the circuit in (3), we find that its expression is

/a2 /"<*!

wdz + a (I -a) I wdz,
0 J 0

and these two values must be equal.

But the integrals denoted by the same symbols are not the same in the two cases ; the

function I * wdz is different in the second value of J from that in the first, for the deforma-

Jo

tion of path necessary to change from the one to the other passes over the branch-point az.
In fact, the equality of the two values of / really determines the value of the integral for
the loop Oal in (3).

And, in general, equations thus obtained by varied deformations do not give relations
among loop-integrals but define the values of those loop-integrals for the deformed paths.

We therefore take that deformation of the circuit into loops which gives the simplest
path. Usually the path is changed into a group of loops round the branch-points as they
occur, taken in order in a trigonometrically positive direction.

The value of the integral round a circuit, equivalent to any number of loops, is obvious.

Ex. 2. To find the value of $wdz, taken round a simple curve which includes all the
branch-points of w and all the infinities.

104.] OF PERIODICITY OF INTEGRALS 191

If z = oo be a branch-point or an infinity, then all the branch-points and all the
infinities of w lie on what is usually regarded as the exterior of the curve, or the curve
may in one sense be said to exclude all these points. The integral round the curve is then
the integral of a function round a curve, such that over the area included by it the
function is uniform, finite and continuous ; hence the integral is zero.

If 0 = 00 be neither a branch-point nor an infinity, the curve can be deformed until it is
a circle, centre the origin and of very great radius. If then the limit of zw, when \z is
infinitely great, be zero, the value of the integral again is zero, by II., § 24.

Another method of considering the integral, is to use Neumann's sphere for the
representation of the variable. Any simple closed curve divides the area of the sphere
into two parts ; when the curve is defined as above, one of those parts is such that the
function is uniform, finite and continuous throughout and therefore its integral round the
curve, regarded as the boundary of that part, is zero. (See Corollary III., § 90.)

Ex. 3. To find the general value of J(l-22)~*cfe. The function to be integrated is
two-valued: the two values interchange round each of the branch-points ±1, which are
the only branch-points of the function.

Let / be the value of the integral for a loop from the origin round +1, beginning with
the branch which has the value +1 at the origin ; and let /' be the corresponding value
for the loop from the origin round - 1, beginning with the same branch. Then, by Ex. 1,

/= 2 P (1 - z*T*dz, /' = 2 f"1 (1 - z2)"* dz

= -/,

the last equality being easily obtained by changing variables.

Now consider the integral when taken round a circle, centre the origin and of indefinitely
great radius R ; then by § 24, II., if the limit of zw for z= QO be k, the value of \wdz round

this circle is 2iri&. In the present case w = (l- 22)~^ so that the limit of zw is + ^ ; hence

J(l-22r^2 = 27T,

the integral being taken round the circle. But since a description of the circle restores the

initial value, it can be deformed into the two loops from 0 O'

to A and from 0 to A'. The value round the first is /; and ^ r > ^

the branch with which the second begins to be described has

the value — 1 at the origin, so that the consequent value round *1S- ^"-

the second is — /' ; hence

7-/' = 2»r*

and therefore

verifying the ordinary result that

when the integral is taken along a straight line.

To find the general value of u for any path of variation between 0 and z, we proceed as
follows. Let Q be any circuit which restores the initial branch of (l-z2)~^. Then by
§ 100, Corollary II., Q may be composed of

(i) a set of double circuits round + 1, say m',
(ii) a set of double circuits round - 1, say m",
and (iii) a set of circuits round + 1 and - 1 ;

* It is interesting to obtain this equation when O' is taken as the initial point, instead of 0.

192 EXAMPLES OF PERIODICITY [104.

and these may come in any order and each may be described in either direction. Now for
a double circuit positively described, the value of the integral for the first description is /
and for the second description, which begins with the branch —(1 — z2)~^, it is — /; hence
for the double circuit it is zero when positively described, and therefore it is zero also when
negatively described. Hence each of the TO' double circuits yields zero as its nett contribu
tion to the integral.

Similarly, each of the m" double circuits round - 1 yields zero as its nett contribution
to the integral.

For a circuit round + 1 and - 1 described positively, the value of the integral has just
been proved to be /-/', and therefore when described negatively it is /'-/. Hence if
there be n^ positive descriptions and n2 negative descriptions, the nett contribution of all
these circuits to the value of the integral is (n± — n^) (I - 1'), that is, 2nir where n is an
inteer.

Hence the complete value for the circuit Q i

Now any path from 0 to z can be resolved into a circuit Q, which restores the initial
branch of (1 — 22)~ , chosen to have the value
+ 1 at the origin, and either (i) a straight
line Oz ;

or (ii) the path OACz, viz., a loop round
+ 1 and the line Oz ;

or (iii) the path OA'Cz, viz., a loop round
- 1 and the line Oz.

Let u denote the value for the line Oz, so that

u= f* (!-#)-* dk.

J o

Hence, for case (i), the general value of the integral is

2W7T + U.

For the path OA Cz, the value is 7 for the loop OAC, and is ( — u) for the line Cz, the
negative sign occurring because, after the loop, the branch of the function for integra
tion along the line is —(1 — 22)~5 ; this value is I—u, that is, it is TT — U. Hence, for case
(ii), the value of the integral is

— U.

For the path OA'Cz, the value is similarly found to be - TT - u ; and therefore, for case (iii),
the value of the integral is

2?wr — ir-u.

If /(z) denote the general value of the integral, we have either

Or /(Z) = (2TO+1)7T-W,

where n and m are any integers, so that/ (z) is a function with two infinite series of values.
Lastly, if z = $($) be the inverse oif(z} = 6, then the relation between u and z given by

can be represented in the form
and

104.] OF INTEGRALS 193

both equations being necessary for the full representation. Evidently z is a simply -periodic
function of u, the period being 2?r ; and from the definition it is easily seen to be an odd
function.

Let y = (\ -z2)—x (u\ so that y is an even function of u ; from the consideration of the
various paths from 0 to 2, it is easy to prove that

Ex. 4. To find the general value of f{(l-j*)(l-IM)}~*dk It will be convenient
(following Jordan *) to regard this integral as a special case of

Z= \{(z -a)(z- b) (z -c}(z- d)}~* dz = \wdz.

The two-valued function to be integrated has a, 6, c, d (but not oo ) as the complete
system of branch-points ; and the two values interchange at each of them. We proceed as
in the last example, omitting mere re-statements of reasons there given that are applicable
also in the present example.

Any circuit Q, which restores an initial branch of w, can be made up of
(i) sets of double circuits round each of the branch-points,
and (ii) sets of circuits round any two of the branch -points.
The value of \wdz for a loop from the origin to a branch-point k (where k = a,b, c, or d) is

2 I wdz ;

J o
and this may be denoted by K, where K=A, B, C, or D.

The value of the integral for a double circuit round a branch-point is zero. Hence the
amount contributed to the value of the integral by all the sets in (i) as this part of
Q is zero.

The value of the integral for a circuit round a and b taken positively is A - B ; for one
round b and c is B- C ; for one round c and d is C-D; for one round a and c is A - C,
which is the sum of A - B and B-C; and similarly for circuits round a and d and round
b and d. There are therefore three distinct values, say A-B, B-C, C-D, the values
for circuits round a and b, b and c, c and d respectively ; the values for circuits round any
other pair can be expressed linearly in terms of these values. Suppose then that the part
of Q represented by (ii), when thus resolved, is the nett equivalent of the description of m'
circuits round a and b, of n' circuits round b and c, and of I' circuits round c and d. Then
the value of the integral contributed by this part of Q is

• which is therefore the whole value of the integral for Q.

But the values of A, £, C, D are not independent f. Let a circle with centre the origin
and very great radius be drawn ; then since the limit of zw for |s| = oo is zero and since
2= cc is not a branch-point, the value of \wdz round this circle is zero (Ex. 2). The circle
can be deformed into four loops round a, b, c, d respectively in order ; and therefore the
value of the integral is A - B + C- D, that is,

Hence the value of the integral for the circuit fl is

where m and n denote m' - 1' and n' - 1' respectively.
* Cours d' Analyse, t. ii, p. 343.

t For a purely analytical proof of the following relation, see Greenhill's Elliptic Functions
Chapter II.

F- 13

194 PERIODICITY [104.

Now any path from the origin to z can be resolved into Q, together with either
(i) a straight line from 0 to z,

or (ii) a loop round a and then a straight line to z.

It might appear that another resolution would be given by a combination of Q with, say, a
loop round b and then a straight line to z ; but it is resoluble into the second of the above
combinations. For at C, after the description of the loop B , introduce a double description
of the loop A, which adds nothing to the value of the integral and does not in the end
affect the branch of w at C ; then the new path can be regarded as made up of (a) the
circuit constituted by the loop round b and the first loop round a, (/3) the second loop round
a, which begins with the initial branch of w, followed by a straight path to z. Of these
(a) can be absorbed into G, and (/3) is the same as (ii) ; hence the path is not essentially
new. Similarly for the other points.

Let u denote the value of the integral with a straight path from 0 to z; then the
whole value of the integral for the combination of Q with (i) is of the form

For the combination of O with (ii), the value of the integral for the part (ii) of
the path is J, for the loop round a, +(-«), for the straight path which, owing to the
description of the loop round a, begins with - w ; hence the whole value of the integral is
of the form

Hence, if / (z) denote the general value of the integral, it has two systems of values, each
containing a doubly -infinite number of terms; and, if z = <j>(u) denote the inverse of
u = f (z\ we have

= 0 {m (A-B} + n(B-C)+A - u},

where m and n are any integers. Evidently z is a doubly-periodic function of u, with
periods A-B and B-C.

Ex. 5. The case of the foregoing integral which most frequently occurs is the elliptic
integral in the form used by Legendre and Jacobi, viz. :

u = J{(1 - z2) (1 - kW)}-*dz = \wdz,
where k is real. The branch-points of the function to be integrated are 1, -1, ^

and -L and the values of the integral for the corresponding loops from the origin are

A/

A
2 I wdz,

J o

r-i ri

2 I wdz— -2 I wdz,
Jo /•

I wdz,
''

and

Now the values for the loops are connected by the equation

* The value for a loop round b and then a straight line to z, just considered, is B - u

= -(A-B) +
being the value in the text with m changed to m - 1.

104.]

OF ELLIPTIC INTEGRALS

195

and so it will be convenient that, as all the points lie on the axis of real variables, we
arrange the order of the loops so that this relation is identically satisfied. Otherwise,
the relation will, after Ex. 1, be a definition of the paths of integration chosen for the
loops.

Among the methods of arrangement, which secure the identical satisfaction of the

Fig. 28.

relation, the two in the figure* are the simplest, the curved lines being taken straight in
the limit ; for, by the first arrangement when k < 1, we have

and, by the second when £ > 1, we have

both of which are identically satisfied. We may therefore take either of them ; let the
former be adopted.

The periods are A-B, B-C, (and C-D, which is equal to B-A\ and any linear
combination of these is a period: we shall take A - B, and B-D. The latter, B-D,
is equal to

n r-i

2 / wdz -2 I wdz,

Jo Jo

which, being denoted by 4/f, gives

4J5T=4 /

JO{(1-22)(1_£222)}4

as one period. The former, A-B, is equal to

2 I wdz -2 I wdz,
Jo Jo

i

/ wdz;

/k

1|(1-

which is 2

this, being denoted by 2iK', gives

dz

dz'

where £'2 + £2=l and the relation between the variables of the integrals is

i

Hence the periods of the integral are 4K and ZiK'. Moreover, A is 2 I" wdz, which i

i J»

2 / wdz + 2 I wdz =
Jo J i

Hence the general value of f* {(I - z*) (I -

* Jordan, Cours d' 'Analyse, t. ii, p. 356.

13—2

196 PERIODICITY [104.

or

that is, 2K-u + 4mK+2niK',

where u is the integral taken from 0 to z along an assigned path, often taken to be
a straight line ; so that there are two systems of values for the integral, each containing
a doubly -infinite number of terms.

If z be denoted by $ (u) — evidently, from the integral definition, an odd function
of u — , then

so that z is a doubly-periodic function of u, the periods being 4A and 2iK'.

Now consider the function ^ = (1 -zrf. A 2-path round T does not affect ^ by way of
change, provided the curve does not include the point 1 ; hence, if zt = x (u), we have

But a z- path round the point 1 does change % into —z1; so that

X («)--* («+**}

Hence x (u\ which is an even function, has two periods, viz., 4AT and 2A' + 2i'A", whence

x(u) = x(u + 4mK+ 2nK+ 2niK').
Similarly, taking z2 = (l -Fs2)* = -f (u), it is easy to see that

so that ^ (u), which is an even function, has two periods, viz., 2 A' and 4iK' ; whence

= u

The functions <£ (u), x (u\ ^ (M) are of course sn w> cn '""> dn M respectively.
Ex. 6. To find the general value of the integral

The function to be integrated has e^ e2, e3, and co for its branch-points; and for
paths round each of them the two branches interchange.

A circuit G which restores the initial branch of the function to be integrated can
be resolved into : —

(i) Sets of double circuits round each of the branch-points alone : as before, the

value of the integral for each of these double circuits is zero.
(ii) Sets of circuits, each enclosing two of the branch-points : it is convenient to
retain circuits including oo and en oo and e.2, oc and e3, the other three
combinations being reducible to these.
The values of the integral for these three retained are respectively

E! = 2 f (4 (z - ej (z - e2} (z - e^dz = 2«1 ,
J «i

Ez=2 I {4(2-e1)(2-e2)(s-e3)}~ick=2a>2,
J 62

3 J ea

* The choice of o> for the upper limit is made on a ground which will subsequently be
considered, viz., that, when the integral is zero, z is infinite.

104.] OF ELLIPTIC INTEGRALS 197

and therefore the value of the integral for the circuit O is of the form

But E^ K2, E3 are not linearly independent. The integral of the function round any

curve in the finite part of the plane, which does not

include el5 e<2 or e3 within its boundary, is zero, by Ex. 2;

and this curve can be deformed to the shape in the figure,

until it becomes infinitely large, without changing the

value of the integral.

Since the limit of zw for \z\ = 00 is zero, the value of
the integral from oo ' to oo is zero, by § 24, II. ; and if the
description begin with a branch w, the branch at oo is -w.
The rest of the integral consists of the sum of the values Fig. 29.

round the loops, which is

because a path round a loop changes the branch of w and the last branch after describing the
loop round e3 is +w at GO', the proper value (§ 90, in). Hence, as the whole integral
is zero, we have

or say E2 =

Thus the value of the integral for any circuit Q, which restores the initial branch of w, can
be expressed in any of the equivalent forms mE^ n E3, m'E^n'E^ m"E2 + ri'Ez, where
the m's and ris, are integers.

Now any path from co to z can be resolved into a circuit fl, which restores at oo the
initial branch of w, combined with either

(i) a straight path from oo to 2,

or (ii) a loop between oo and e1} together with a straight path from oo to z.
' (The apparently distinct alternatives, of a loop between oo and e2 together with a straight
: path from oo to z and of a similar path round ea, are inclusible in the second alternative
above ; the reasons are similar to those in Ex. 5.)

fx

If u denote j ^ {^(z-ej (z-e2) (z-e3}}~*dz when the integral is taken in a straight

, line, then the value of the integral for part (i) of a path is u; and the value of the
1 integral for part (ii) of a path is El - u, the initial branch in each case for these parts being
. the initial branch of w for the whole path. Hence the most general value of the integral
for any path is

+ 2no>3 + u,

or

the two being evidently included in the form

2mo>1 + 2n(,)3±u.
If, then, we denote by z = ft>(u) the relation which is inverse to

we

In the same way as in the preceding example, it follows that

where ^ («) is - {4 (z - e^ (z - e2) (z - e3)}*.

198 SIMPLE PERIODICITY [104

The foregoing simple examples are sufficient illustrations of the multi
plicity of value of an integral of a uniform function or of a multiform
function, when branch-points or discontinuities occur in the part of the plane
in which the path of integration lies. They also shew one of the modes in
which singly-periodic and doubly-periodic functions arise, the periodicity
consisting in the addition of arithmetical multiples of constant quantities
to the argument. And it is to be noted that, as only a single value of z
is used in the integration, so only a single value of z occurs in the
inversion ; that is, the functions just obtained are uniform functions of their
variables. To the properties of such periodic functions we shall return in the
succeeding chapters.

105. We proceed to the theory of uniform periodic functions, some
special examples of which have just been considered ; and limitation will
be made here to periodicity of the linear additive type, which is only a very
special form of periodicity.

A function f(z) is said to be periodic when there is a quantity &> such
that the equation

/(* + »)=/(*)

is an identity for all values of z. Then/0 + nw) =f(z), where n is any
integer positive or negative; and it is assumed that &> is the smallest
quantity for which the equation holds, that is, that no submultiple of &> will
satisfy the equation. The quantity u> is called a period of the function.

A function is said to be simply-periodic when there is only a single
period : to be doubly-periodic when there are two periods ; and so on, the
periodicity being for the present limited to additive modification of the
argument.

It is convenient to have a graphical representation of the periodicity of a
function.

(i) For simply-periodic functions, we
take a series of points 0, A1} A2,...,
A-i, ^4_2,... representing 0, w, 2o>, ... ,
— <», — 2&>, . . . ; and through these points
we draw a series of parallel lines, dividing
the plane into bands. Let P be any
point z in the band between the lines
through 0 and through A^\ through P
draw a line parallel to OAl and measure

each equal to OA^ then all the points /

P1} P2, ... , P_i, P-2, ... are represented

by z + nco for positive and negative integral values of n. But/ (2 + &»)=/(•*)]

and therefore the value of the function at a point Pn in any of the bands is

105.]

DOUBLE PERIODICITY

199

the same as the value at P. Moreover to a point in any of the bands there
corresponds a point in any other of the bands ; and therefore, owing to the
periodic resumption of the value at the points corresponding to each point P,
it is sufficient to consider the variation of the function for points within one
band, say the band between the lines through 0 and through AI. A point P
within the band is sometimes called irreducible, the corresponding points P
in the other bands reducible.

If it were convenient, the boundary lines of the bands could be taken
through points other than Al} A2, ... ; for example, through points (m + |) &>
for positive and negative integral values of ra. Moreover, they need not be
straight lines. The essential feature of the graphic representation is the
division of the plane into bands.

(ii) For doubly-periodic functions a similar method is adopted. Let &>
and co' be the two periods of such a
function /(#), so that

/<«.+»)»/(*)-/(•+ <0;

then f(z + nw + n'w) =f(z),
where n and n' are any integers positive
or negative.

For graphic purposes, we take points
0, A-L, A2, ..., A^i, A_2, ... representing
0, ft), 2&), . . . , — to, — 2(w, . . . ; and we take
another series 0, B1} B2, . . . , B_1} B_2, . . .
representing 0, &)', 2&/, . . . , — ft/, — 2ft/, . . . ;
through the points A we draw lines
parallel to the line of points B, and
through the points B we draw lines
parallel to the line of points A. The intersection of the lines through An
and Bn> is evidently the point n&> + w'&>', that is, the angular points of the
parallelograms into which the plane is divided represent the points nco + n'w
for the values of n and n'.

Let P be any point z in the parallelogram OAfi-JS^ ; on lines through P,
parallel to the sides of the parallelogram, take points Q1} Q2, ... , Q_lt Q_2, ...
such that PQl = QiQ2= ... = ft) and points Rlt R2, ... , R_lt R_2, ... such that
PRl = R^ — . . . — to' ; and through these new points draw lines parallel to
the sides of the parallelogram. Then the variables of the points in which
these lines intersect are all represented by z + mw + mV for positive and nega
tive integral values of m and m' ; and the point represented by z + m^ + m'a)'
is situated in the parallelogram, the angular points of which are mw 4 mot',
(m + 1) &) + mw, mco + (mf + 1) ft)', and (m -f 1) &) + (m + 1) ft/, exactly as P
is situated in OA^C^. But

/ (z + m^ + Wj V) = / (z\

Fig. 31.

200 RATIO OF THE PERIODS [105.

and therefore the value of the function at such a point is the same as the
value at P. Since the parallelograms are all equal and similarly situated.
to any point in any of them there corresponds a point in OA^G^B^; and the
value of the function at the two points is the same. Hence it is sufficient to
consider the variation of the function for points within one parallelogram, say,
that which has 0, &>, o) + «', &>' for its angular points. A point P within
this parallelogram is sometimes called irreducible, the corresponding points
within the other parallelograms reducible to P ; the whole aggregate of the
points thus reducible to any one are called homologous points. And the
parallelogram to which the reduction is made is called the parallelogram of
periods.

As in the case of simply-periodic functions, it may prove convenient to
choose the position of the fundamental parallelogram so that the origin is
not on its boundary ; thus it might be the parallelogram the middle points of
whose sides are + £&>, + ^co'.

106. In the preceding representation it has been assumed that the line
of points A is different in direction from the line of points B. If &> = u + iv
and to' = u'+iv', this assumption implies that v'/u' is unequal to v/u, and
therefore that the real part of a>'/ia> does not vanish. The justification of
this assumption is established by the proposition, due to Jacobi * : —

The ratio of the periods of a uniform doubly -periodic function cannot be
real.

Let/ (2) be a function, having CD and CD' as its periods. If the ratio w'/to
be real, it must be either commensurable or incommensurable.

If it be commensurable, let it be equal to n'/n, where n and n' are
integers, neither of which is unity owing to the definition of the periods CD
and 6Dj.

Let n'/n be developed as a continued fraction, and let m'fm be the last
convergent before n'jn, where m and mf are integers. Then

n' m _ 1

n m mn'
that is, mn' - m'n = 1,

, 1 / . U> , . .. CD

so that mco ~ mco = -(mn~ run ) = - .

n x n

Therefore f(z) =f(z + m'co ~ mco'),

since m and m' are integers ; so that

-, , ~( co\

/(*)-/(' -i- s).

contravening the definition of CD as a period, viz., that no submultiple of co is a
period. Hence the ratio of the periods is not a commensurable real quantity.

* Ges. Werke, t. ii, pp. 25, 26.

106.] OF A UNIFORM DOUBLY-PERIODIC FUNCTION 201

If it be incommensurable, we express oj'/aj as a continued fraction. Let
p/q and p'/q' be two consecutive convergents : their values are separated by
the value of &>'/&>, so that we may write

v~q+ \q'~q)'

where 1 > h > 0.

Now pq <- p'q — 1, so that

- = P + —
o> q qq

where e is real and |e < 1 ; hence

, e

qa) —pa) = —, &>.

Therefore f(z) =f(z + qw — pa),

since p and q are integers ; so that

Now since &>'/&> is incommensurable, the continued fraction is unending. We
therefore take an advanced convergent, so that q' is very large. Then €- &> is

a very small quantity and z + - &> is a point infinitesimally near to z, that

is, the function / (V), under the present hypothesis, resumes its value at a
point infinitesimally near to z. Passing along the line joining these two
points infinitesimally near another, we should have / (z) constant along a
line and therefore (§ 37) constant everywhere ; it would thus cease to be a
varying function.

The ratio of the periods is thus not an incommensurable real quantity.

We therefore infer Jacobi's theorem that the ratio of the periods cannot
be real. In general, the ratio is a complex quantity ; it may, however, be a
pure imaginary*.

COROLLARY. If a uniform function have two periods wl and &>2 such that
a relation

mlwl + ra2G>2 = 0

exists for integral values of m1 and ?n2, the function is only simply-periodic.
And such a relation cannot exist between two periods of a simply-periodic
function, if m^ and ra2 be real and incommensurable ; for then the function
would be constant.

* It was proved, in Ex. 5 and Ex. 6 of § 104, that certain uniform functions are doubly-periodic.
A direct proof, that the ratio of the distinct periods of the functions there obtained is not a real
quantity, is given by Falk, Acta Math., t. vii, (1885), pp. 197—200, and by Pringsheim, Math.
Ann., t. xxvii, (1886), pp. 151—157.

202 UNIFORM [106.

Similarly, if a uniform function have three periods &>1; a>.2> o>3, connected

by two relations

.. = 0,

n1o)1 + n2a)2 + n3a)3 = 0,

where the coefficients m and n are integers, then the function is only simply-
periodic.

107. The two following propositions, also due to Jacobi*, are important
in the theory of uniform periodic functions of a single variable : —

If a uniform function have three periods w^, «2, MS such that a relation

m^i + m.2&>2 + m3w3 = 0

is satisfied for integral values ofmlt w2, m3, then the function is only a doubly-
periodic function.

What has to be proved, in order to establish this proposition, is that two
periods exist of which wl, &>2, &>3 are integral multiple combinations.

Evidently we may assume that m^, ra2, m3 have no common factor: let /
be the common factor (if any) of m.2 and m3, which is prime to m^. Then
since

and the right-hand side is an integral combination of periods, it follows that
riod.

is a fraction in its lowest terms. Change it into a continued

-~ &>! is a period.

fraction and let ^ be the last convergent before the proper value ; then
2

1

so that <l~f~P=±^f-

But o>! is a period and ^ft)! is a period; therefore q —^ Wj — pwi is a period,

or &>!// is a period, = to/ say.

Let ra2//= m2', m3/f= m/, so that m1&V + m2'&>2 + ??i3'&)3 = 0. Change

fy»

m.2'/m3 into a continued fraction, taking - to be the last convergent before the

proper value, so that

m/ r _ 1

/ i

s sms

* Ges. Werke, t. ii, pp. 27—32.

107.] DOUBLY-PERIODIC FUNCTIONS 203

Then r&>2 + sco., being an integral combination of periods, is a period. But
± &>2 = &)2 (sm2r — rm3)

= — ra>.2m3 — s (m^ + w3'&>3)

= — m^sw-i' - ma' (r&>2 + su>3) ;
also + ft)3 = &)3 (sm/ — rm3)

— sm2'o)3 + r (mjO)/

and o>! =/&)/.

Hence two periods &>/ and r<u2 + s&>3 exist of which co1} co2, &>3 are integral
multiple combinations ; and therefore all the periods are equivalent to &>/ and
r&>2 + so)3, that is, the function is only doubly-periodic.

COROLLARY. If a function have four periods <ul3 &>2, cos, &>4 connected by
two relations

m1o)1 + m2o)2 + wi3ft)3 + ra4&>4 = 0,

72J60J + W20)2 + W3ft)3 + W4«04 = 0,

where the coefficients m and w are integers, the function is only doubly-
periodic.

108. If a uniform function of one variable have three periods a)l, w.,, &>3,
then a relation of the form

m1o)l + w?2to2 + in3(i)3 = 0
must be satisfied for some integral values ofml} m2, ms.

Let a)r = ar + i@r, for r = 1, 2, 3 ; in consequence of § 106, we shall assume
that no one of the ratios of twj, <w2, w3 in pairs is real, for, otherwise, either
the three periods reduce to two immediately, or the function is a constant.
Then, determining two quantities A, and fj, by the equations

so that X and //, are real quantities and neither zero nor infinity, we have

for real values of X and p.

Then, first, if either X or fj. be commensurable, the other is also commen
surable. Let X = a/6, where a and b are integers ; then

= bo)3 — aa)},

so that fyu,&>2 is a period. Now, if b/j, be not commensurable, change it into a
continued fraction, and let p/q, p'/q be two consecutive convcrgents, so that,
as in § 106,

/ P , x
bfji,=^+ —,,

q qq

204 TRIPLY-PERIODIC UNIFORM [108.

where 1 > x > — 1. Then - &>., + -— ? is a period, and so is <w2 ; hence

q qq

'P~ ^x

IT

is a period, that is, - <a2 is a period. We may take q indefinitely large, and

then the function has an infinitesimal quantity for a period, that is, it would
be a constant under the hypothesis. Hence &/* (and therefore /*) cannot be
incommensurable, if X be commensurable; and thus X and //. are simul
taneously commensurable or simultaneously incommensurable.

CL G

If X and fj, be simultaneously commensurable, let X = j- , p = -^ , so that

a c

&)3 = r &>! + -jG>2.

o a

and therefore 6rfto3 = ac^ + bca)2,

a relation of the kind required.

If X and //. be simultaneously incommensurable, express A, as a continued
fraction ; then by taking any convergent r/s, we have

r _ x
*=*'

/Yt

where 1 > x > — 1, so that s\ — r=-:

s

by taking the convergent sufficiently advanced the right-hand side can be
made infinitesimal.

Let i\ be the nearest integer to the value of s/j,, so that, if

we have A numerically less than ^. Then

x

sat-, — ra>1 — r1w2 = — a)1 + Aw.,,
s

fp

and the quantity - Wj can be made so small as to be negligible. Hence

S

integers r, rlt s can be chosen so as to give a new period &>/(= A&>2), such
that | &)/ < \ &)2 .

We now take wl, &>2', &>3: they will be connected by a relation of the form

0>3 =X'(W1 +yLt/G)2/,

and X' and // must be incommensurable : for otherwise the substitution for
to/ of its value just obtained would lead to a relation among a>l) &>o, &>3 that
would imply commensurability of X and of p.

Proceeding just as before, we may similarly obtain a new period &>2" such
that <o2" < \ ! mz I and so on in succession. Hence we shall obtain, after n

108.] FUNCTIONS DO NOT EXIST 205

such processes, a period co2(w) such that |&)2(n)| < ^ a>*\, so that by making n

z

sufficiently large we shall ultimately obtain a period less than any assigned
quantity. Let such period be to ; then

/(*+«)-/(*),

and so for points along the co-line we have an infinite number close together
at which the function is unaltered in value. The function, being uniform,
must in that case be constant.

It thus appears that, if A. and /j, be simultaneously incommensurable, the
function is a constant. Hence the only tenable result is that A. and //. are
simultaneously commensurable, and then there is a period-equation of the

form

m^w^ + m.2o)2 + m3o)s = 0,

where m1, w2, m3 are integers.

The foregoing proof is substantially due to Jacobi (I.e.). The result can
be obtained from geometrical considerations by shewing that the infinite
number of points, at which the function resumes its value, along a line
through z parallel to the two-line will, unless the condition be satisfied, reduce
to an infinite number of points in the a)1, &)2 parallelogram which will form
either a continuous line or a continuous area, in either of which cases the
function would be a constant. But, if the condition be satisfied, then the
points along the line through z reduce to only a finite number of points.

COROLLARY I. Uniform functions of a single variable cannot have three
independent periods ; in other words, triply -periodic uniform functions of a
single variable do not exist* ; and, a fortiori, uniform functions of a single
variable with a number of independent periods greater than two do not exist.

But functions involving more than one variable can have more than two
periods, e.g., Abelian transcendents ; and a function of one variable, having
more than two periods, is not uniform.

COROLLARY II. All the periods of a uniform periodic function of a single
variable reduce either to integral multiples of one period or to linear combina
tions of integral multiples of two periods whose ratio is not a real quantity.

109. It is desirable to have the parallelogram, in which a doubly-
periodic function is considered, as small as possible. If in the parallelogram
(supposed, for convenience, to have the origin for an angular point) there be
a point a)" such that

/(* + »")=/(*)
for all values of z, then the parallelogram can be replaced by another.

* This theorem is also due to Jacobi, (I.e., p. 202, note).

206 FUNDAMENTAL PARALLELOGRAM [109.

It is evident that co" is a period of the function ; hence (§ 108) we must

have

co" = Aco + /AW' ;

and both X and /JL, which are commensurable quantities, are less than unity
since the point is within the parallelogram. Moreover, co -f co' — <»", which
is equal to (1 — A,) co + (1 — /"•)&>', is another point within the parallelogram;
and

/(* + » + «'-«")«/(*),

since co, co', co" are periods. Thus there cannot be a single such point, unless

X = \ = p.

But the number of such points within the parallelogram must be finite ;
if there were an infinite number, they would form a continuous line or a
continuous area where the uniform function had an unvarying value, and
consequently (§ 37) the function would have a constant value everywhere.

To construct a new parallelogram when all the points are known, we first
choose the series of points parallel to the co-line through the origin 0, and of
that series we choose the point nearest 0, say Al. We similarly choose the
point, nearest the origin, of the series of points parallel to the co-line and
nearest to it after the series that includes Al} say Bl : we take OA1} OB1 as
adjacent sides of the parallelogram, and these lines as the vectorial repre
sentations of the periods. No point lies within this parallelogram where the
function has the same value as at 0 ; hence the angular points of the original
parallelograms coincide with angular points of the new parallelograms.

When a parallelogram has thus been obtained, containing no internal
point fl such that the function can satisfy the equation

for all values of z, it is called a fundamental, or a primitive, parallelogram, :
and the parallelogram of reference in subsequent investigations will be
assumed to be of a fundamental character.

But a fundamental parallelogram is not unique.

Let co and co' be the periods for a given fundamental parallelogram, so
that every other period co" is of the form Aco + //-co', where A, and /* are
integers. Take any four integers a, b, c, d such that ad — lc=±l, as may
be done in an infinite variety of ways ; and adopt two new periods coj and co2,
such that

&>! = aco + bo)', co2 = ceo + d(o'.

Then the parallelogram with coj and co2 for adjacent sides is fundamental.
For we have

+ eo = do)1 — ba>2, + co' = — ccox + aco2,

and therefore any period co"
= A.CO + /uco'
= (\d - fie) wl + (— \b + fj.a) eo2, save as to signs of A, and /z.

109.] OF PERIODS 207

The coefficients of o^ and &)2 are integers, that is, the point <w" lies outside
the new parallelogram of reference; there is therefore no point in it such that

/(* + *>")=/(*),
and hence the parallelogram is fundamental.

COROLLARY. The aggregate of the angular points in one division of the
plane into fundamental parallelograms coincides with their aggregate in
any other division into fundamental parallelograms ; and all fundamental
parallelograms for a given function are of the same area.

The method suggested above for the construction of a fundamental parallelogram is
geometrical, and it assumes a knowledge of all the points w" within a given parallelogram
for which the equation/ (z -f «")=/ (z) is satisfied.

Such a point o>3 within the o^, o>2 parallelogram is given by

nil m2

<Bo= - (Bi -\ -- 0>9,

' m3 J m3 2

where »&1} m2, m3 are integers. We may assume that no two of these three integers
have a common factor; were it otherwise, say for m^ and wi2, then, as in § 107, a
submultiple of o>3 would be a period — a result which may be considered as excluded.
Evidently all the points in the parallelogram are the reduced points homologous with
w3, 2o>3, ...... , (m3 — 1)«3; when these are obtained, the geometrical construction is

possible.

The following is a simple and practicable analytical method for the construction.

Change w^/rag and mz/m3 into continued fractions; and let p/q and r/s be the
last convergents before the respective proper values, so that

mx p e m2 r f'

m3 q gm3' m3 s sm3'
where e and e' are each of them +1. Let

m"> n , M ml j , ^

q — =d + — , s^ = $+ — ,

m3 m3 m3 m3

where X and p, are taken to be less than m3, but they do not vanish because q and s are
less than m3. Then

2'eo3-^w1-(9o)2 = — (/*a>2 + f»i), *a>3-ro>2 -<£<•>! = — (Xa^ + e'tOjj) ;

U vn II io

the left-hand sides are periods, say Qx and O2 respectively, and since /u + e is not >m3 and
X + e' is not >m3, the points Q.l and Q2 determine a parallelogram smaller than the initial
parallelogram.

Thus
are equations defining new periods Qly Q2. Moreover

, . X m-. p 65 a m9 r t'o

4>-\ -- = s-^=s*-+ -, 0 + -f^ = n —? = «- + -L :

m3 m3 q qms m3 2 ms * s sm3

so that, multiplying the right-hand sides together and likewise the left-hand sides, we
at once see that X/i-ee' is divisible by ms if it be not zero: let

X/i — ee' = wi3A.

Then, as X and p are less than m3, they are greater than A; and they are prime to it,
because ee' is +1.

208 MULTIPLE [109.

Hence we have Aa>j = ^Q2- t'Ql, Aa>2 = XQ1- eiV

Since X and /u are both greater than A, let

X = X1A + X', /x = /i1A + //,
where X' and /x' are <A. Then X'/*'— «' ig divisible by A if it be not zero, say

X'p - ee = AA' ;
then X' and p.' are >A' and are prime to it. And now

A (wj — /^iO2) = /x'Q2 ~ e'^i > A (W2 "~ ^1^1) = ^-'QI ~ f®2 i

and therefore, if (a1 — /^G^Qg, <B2-X1Q1 = Q4, which are periods, we have

With Q3 and Q4 we can construct a parallelogram smaller than that constructed
with Qj and Q2.

We now have A'Q1 = fG3+//G4, A'Q.j=X'Q3 + e'fl4,

that is, equations of the same form as before. We proceed thus in successive stages :
each quantity A thus obtained is distinctly less than the preceding A, and so finally we
shall reach a stage when the succeeding A would be unity, that is, the solution of the pair
of equations then leads to periods that determine a fundamental parallelogram. It
is not difficult to prove that a>lt o>2, o>3 are combinations of integral multiples of these
periods.

If one of the quantities, such as X'/x'-ee', be zero, then X'=/x' = l, e = e'= ±1 ; and then
Q3 and O4 are identical. If e = e' = + 1, then AQ3 = Q2 - Qj , and the fundamental parallelo
gram is determined by

<V = QI + - (Q2 - %), G4' = Q2 - 1 (Q2 - Qt)-

If f = f = -1, then AQ3 = Q2+O15 so that, as A is not unity in this case, the fundamental
parallelogram is determined by Q2 and Q3.

Ex. If a function be periodic in a>1? a>2, and also in <o3 where

29co = 1 7

periods for a fundamental parallelogram are

QI = Scoj + 3o)2 - 8w3 , Q2' = 3 eoj + 2co2 - 5w3 ,
and the values of a>1} <»2, w3 in terms of O/ and Q2' are

G)2 = 2-1, a>3 = 2 Q.

Further discussion relating to the transformation of periods and of fundamental
parallelograms will be found in Briot and Bouquet's The'orie des fonctions elliptujues,
pp. 234, 235, 268—272.

110. It has been proved that uniform periodic functions of a single
variable cannot have more than two periods, independent in the sense that
their ratio is not a real quantity. If then a function exist, which has two
periods with a real incommensurable ratio or has more than two independent
periods, either it is not uniform or it is a function (whether uniform or multi
form) of more variables than one.

When restriction is made to uniform functions, the only alternative is
that the function should depend on more than one variable.

110.] PERIODICITY 209

In the case when three periods o)l, &>2, &>3 (each of the form a-f t/3) were
assigned, it was proved that the necessary condition for the existence of a
uniform function of a single variable is that finite integers mly m2, m:i can
be found such that

ra2cr2 + m3a3 = 0,

- w3/33 = 0 ;

and that, if these conditions be not satisfied, then finite integers m1} m.2, ms
can be found such that both Sma and 2m/8 become infinitesimally small.

This theorem is purely algebraical, and is only a special case of a more
general theorem as follows :

Let an, or12,..., alt r+1; a21, aw,..., a2>r+1;...; arl, «.«,..., ar,r+i be r sets of
real quantities such that a relation of the form

wia*i + ^2^2 + . • • + nr+l <xsr+i = 0

is not satisfied among any one set. Then finite integers m^,..., mr+1 can be
determined such that each of the sums

j (for 5 = 1, 2,...,r) is an infinitesimally small quantity. And, a fortiori, if
fewer than r sets, each containing r+1 quantities be given, the r+1
integers can be determined so as to lead to the result enunciated ; all that

i is necessary for the purpose being an arbitrary assignment of sets of real

| quantities necessary to make the number of sets equal to r. But the result

! is not true if more than r sets be given.

We shall not give a proof of this general theorem* ; it would follow the
lines of the proof in the limited case, as given in § 108. But the theorem
will be used to indicate how the value of an integral with more than
two periods is affected by the periodicity.

Let / be the value of the integral taken along some assigned path from
an initial point ZQ to a final point z\ and let the periods be &)1} &>2,..., &>r,
(where r > 2), so that the general value is

/ + fftjcoj + m2a)., + . . . + mrwr,

where mlt m2,..., mr are integers. Now if cos = as + i/3s, for s=l, 2,..., r,
when it is divided into its real and its imaginary parts, then finite integers
Wi, n2,..., nr can be determined such that

and n-if.

are both infinitesimal ; and then

2 ns

is infinitesimal. But the addition

of S nscos still gives a value of the integral ; hence the value can be modified

* A proof will be found in Clebsch and Gordan's Theorie der Abel'schen Functioncn, § 38.
F- 14

210 MULTIPLE PERIODICITY [110.

by infinitesimal quantities, and the modification can be repeated indefinitely.
The modifications of the value correspond to modifications of the path from
ZQ to z ; and hence the integral, regarded as depending on a single variable,
can be made, by modifications of the path of the variable, to assume any
value. The integral, in fact, has not a definite value dependent solely
upon the final value of the variable; to make the value definite, the path
by which the variable passes from the lower to the upper limit must be
specified.

It will subsequently (§ 239) be shewn how this limitation is avoided by
making the integral, regarded as a function, depend upon a proper number
of independent variables — the number being greater than unity.

Ex. 1. If F0 be the value of i — -, , (n integral), taken along an assigned path,

Jo (\-znY
and if

P = 2 I1—- ^-j(# real),

then the general value of the integral is

I \ ^ I

n

where q is any integer and mp any positive or negative integer such that 2 mp = 0.

P=I

(Math. Trip. Part II, 1889.)

Ex. 2. Prove that v= I udz, where
J o

is an algebraical function satisfying the equation

and obtain the conditions necessary and sufficient to ensure that

i) = fadz
should be an algebraical function, when u is an algebraical function satisfying an equation

(Liouville, Briot and Bouquet.)

CHAPTER X.

SIMPLY-PERIODIC AND DOUBLY-PERIODIC FUNCTIONS.

111. ONLY a few of the properties of simply-periodic functions will be
given, partly because some of them are connected with Fourier's series the
detailed discussion of which lies beyond our limits, and partly because, as
will shortly be explained, many of them can at once be changed into
properties of uniform non-periodic functions which have already been
considered.

When we use the graphical method of § 105, it is evident that we need
consider the variation of the function within only a single band. Within
that band any function must have at least one infinity, for, if it had not, it
would not have an infinity anywhere in the plane and so would be a constant ;
and it must have at least one zero, for, if it had not, its reciprocal, also a
simply-periodic function, would not have an infinity in the band. The
infinities may, of course, be accidental or essential : their character is repro
duced at the homologous points in all the bands.

For purposes of analytical representation, it is convenient to use a
relation

Ziri

so that, if the point Z in its plane have R and (*)
for polar coordinates,

, „

Z = =— ; log R +

Z7TI

ft).

If we take any point A in the ^-plane and a
corresponding point a in the z-plane, then, as Z
describes a complete circle through A with the
origin as centre, z moves along a line aal} where
di is a + a). A second description of the circle
makes z move from ax to aa, where a2 = ax + &> •

Fig. 32.

and so on in succession.
14—2

212 SIMPLE PERIODICITY [111.

For various descriptions, positive and negative, the point a describes a line,
the inclination of which to the axis of real quantities is the argument of &>.

Instead of making Z describe a circle through A, let us make it describe
a part of the straight line from the origin through A, say from A, where
OA = R, to C, where 00 = R'. Then z describes a line through a perpend
icular to aal} and it moves to c where

Similarly, if any point A' on the former circumference move radially to a
point C at a distance R from the ^-origin, the corresponding z point a'
moves through a distance a'c', parallel and equal to ac : and all the points c'
lie on a line parallel to aa^. Repeated description of a ^-circumference with
the origin as centre makes z describe the whole line cCjCo.

If then a function be simply-periodic in &>, we may conveniently take
any point a, and another point a^ = a + w, through a and a^ draw straight
lines perpendicular to aa1} and then consider the function within this band.
The aggregate of points within this band is obtained by taking

(i) all points along a straight line, perpendicular to a boundary of

the band, as aa^ ;
(ii) the points along all straight lines, which are drawn through the

points of (i) parallel to a boundary of the band.

In (i), the value of z varies from 0 to co in an expression a + z, that is, in
the ^-plane for a given value of R, the angle © varies from 0 to 2?r.

In (ii), the value of log R varies from — oo to +00 in an expression
fi\

. log R + =— w, that is, the radius R must vary from 0 to oo .

2?r

Hence the band in the 0-plane and the whole of the ^-plane are made
equivalent to one another by the transformation

Now let z0 be any special point in the finite part of the band for a given
simply-periodic function, and let Z0 be the corresponding point in the Z-planej
Then for points z in the immediate vicinity of z0 and for points Z which
are consequently in the immediate vicinity of Z0, we have

Ziri

e
to

where | X differs from unity only by an infinitesimal quantity.

111.] FOURIER'S THEOREM 213

If then w, a function of z, be changed into W a function of Z, the following
relations subsist : —

When a point ZQ is a zero of w, the corresponding point ZQ is a zero

of W.

When a point z0 is an accidental singularity of w, the corresponding
point Z0 is an accidental singularity of W.

When a point z0 is an essential singularity of w, the corresponding
point Z0 is an essential singularity of W.

When a point z0 is a branch- point of any order for a function w, the
corresponding point Z0 is a branch-point of the same order for W.

And the converses of these relations also hold.

Since the character of any finite critical point for w is thus unchanged by the
transformation, it is often convenient to change the variable to Z so as to let
the variable range over the whole plane, in which case the theorems already
proved in the preceding chapters are applicable. But special account must
be taken of the point z = oo .

112. We can now apply Laurent's theorem to deduce what is practically

Fourier's series, as follows.

Let f(z) be a simply-periodic function having w as its period, and suppose
that in a portion of the z-plane bounded by any two parallel lines, the inclina
tion of which to the axis of real quantities is equal to the argument of w, the
function is uniform and has no singularities; then, at points within that
portion of the plane, the function can be expressed in the form of a converging

2n-2t

series of positive and of negative integral powers of e "° .

In figure 32, let aa^a^... and cc^... be the two lines which bound the
portion of the plane : the variations of the function will all take place within
that part of the portion of the plane which lies within one of the repre
sentative bands, say within the band bounded by ...ac... and . ..a^...: that is,
we may consider the function within the rectangle acc^a, where it has no
singularities and is uniform.

Now the rectangle acc^a in the 2-plane corresponds to a portion of the
Z-plane which, after the preceding explanation, is bounded by two circles

2iri 2irf

with the origin for common centre and of radii | e w " | and | e u ' ; and the
variations of the function within the rectangle are given by the variations of
a transformed function within the circular ring. The characteristics of the
one function at points in the rectangle are the same as the characteristics of
the other at points in the circular ring : and therefore, from the character
of the assigned function, the transformed function has no singularities and it

214 FOURIER'S THEOREM [112.

is uniform within the circular ring. Hence, by Laurent's Theorem (§ 28),
the transformed function is expressible in the form

a series which converges within the ring : and the value of the coefficient an

is given by

1

tvfj Zn+*

taken along any circle in the ring concentric with the boundaries.

Retransforming to the variable z, the expression for the original function
is

71 = + oo Zrnriz

f(z) = 2 ane~^~ .

71= -00

The series converges for points within the rectangle and therefore, as it
is periodic, it converges within the portion of the plane assigned. And the
value of an is

Zniriz
(?\P *»~ d?

\z) 6 az,

taken along a path which is the equivalent of any circle in the ring concentric
with the boundaries, that is, along any line a'c' perpendicular to the lines
which bound the assigned portion of the plane.

The expression of the function can evidently be changed into the form

Znvi, _

1 r

-±

Wj 7

where the integral is taken along the piece of a line, perpendicular to the
boundaries and intercepted between them.

If one of the boundaries of the portion of the plane be at infinity, (so that
the periodic function has no singularities within one part of the plane), then
the corresponding portion of the ^-plane is either the part within or the part
without a circle, centre the origin, according as the one or the other of the
boundaries is at oo . In the former case, the terms with negative indices
n are absent ; in the latter, the terms with positive indices are absent.

113. On account of the consequences of the relation subsisting between
the variables z and Z, many of the propositions relating to general uniform
functions, as well as of those relating to multiform functions, can be changed,
merely by the transformation of the variables, into propositions relating to
simply-periodic functions. One such proposition occurs in the preceding
section ; the following are a few others, the full development being unnecess
ary here, in consequence of the foregoing remark. * The band of reference
for the simply-periodic functions considered will be supposed to include the

113.] SIMPLY-PERIODIC FUNCTIONS 215

origin : and, when any point is spoken of, it is that one of the series of
homologous points in the plane, which lies in the band.

We know that, if a uniform function of Z have no essential singularity,
then it is a rational algebraical function, which is integral if z = cc be the
only accidental singularity and is meromorphic if there be accidental singu
larities in the finite part of the plane ; and every such function has as many
zeros as it has accidental singularities.

Hence a uniform simply-periodic function with z=cc as its sole essential
singularity has as many zeros as it has infinities in each band of the plane ;
the number of points at which it assumes a given value is equal to the number
of its zeros ; and, if the period be w, the function is a rational algebraical

ZTTIZ

function of e a , which is integral if all the singularities be at an infinite
distance and is meromorphic if some (or all) of them be in a finite part of
the plane. But any number of the zeros and any number of the infinities
may be absorbed in the essential singularity at z = oo .

The simplest function of Z, thus restricted to have the same number of
zeros as of infinities, is one which has a single zero and a single infinity in
the finite part of the plane ; the possession of a single zero and a single infinity
will therefore characterise the most elementary simply-periodic function.
Now, bearing in mind the relation

Zniz

Z=e<*,

the simplest £-pomt to choose for a zero is the origin, so that Z = 1 ; and then
the simplest ^-point to choose for an infinity at a finite distance is \w, (being
half the period), so that Z— — \. The expression of the function in the
Z-plane with 1 for a zero and — 1 for an accidental singularity is

Z~l

and therefore assuming as the most elementary simply-periodic function that
which in the plane has a series of zeros and a series of accidental singularities
all of the first order, the points of the one being midway between those of the
other, its expression is

A

2iriz

e" -I

Zniz

which is a constant multiple of tan — . Since e " is a rational fractional

CD

function of tan — , part of the foregoing theorem can be re-stated as follows: —

If the period of the function be o>, the function is a rational algebraical

function of tan — .

n

216 SIMPLY-PERIODIC [113.

Moreover, in the general theory of uniform functions, it was found con
venient to have a simple element for the construction of products, there
(§ 53) called a primary factor: it was of the type

^Z-u

where the function G ( -~ j could be a constant; and it had only one infinity

and one zero.

Hence for simply-periodic functions we may regard tan — as a typical

primary factor when the number of irreducible zeros and the (equal) number
of irreducible accidental singularities are finite. If these numbers should
tend to an infinite limit, then an exponential factor might have to be

associated with tan — ; and the function in that case might have essential
singularities elsewhere than at z = oo .

114. We can now prove that every uniform function, which has no
essential singularities in the finite part of the plane and is such that all its
accidental singularities and its zeros are arranged in groups equal and
finite in number at equal distances along directions parallel to a given
direction, is a simply-periodic function.

Let to be the common period of the groups of zeros and of singularities :
and let the plane be divided into bands by parallel lines, perpendicular to
any line representing w. Let a, b, ... be the zeros, a, /3, ... the singularities
in any one band.

Take a uniform function </> (z), simply-periodic in <w and having a single
zero and a single singularity in the band : we might take tan — as a value
of <f> (z). Then

is a simply-periodic function having only a single zero, viz., z = a and a single
singularity, viz., z — a. ; for as <f> {z} has only a single zero, there is only a single
point for which (f>(z) = <f) (a), and a single point for which <£ (z) — $ (a). Hence

is a simply-periodic function with all the zeros and with all the infinities of
the given function within the band. But on account of its periodicity it has
all the zeros and all the infinities of the given function over the whole plane ;
hence its quotient by the given function has no zero and no singularity over
the whole plane and therefore it is a constant ; that is, the given function,

114.] FUNCTIONS 217

save as to a constant factor, can be expressed in the foregoing form. It is
thus a simply-periodic function.

This method can evidently be used to construct simply-periodic functions, having
assigned zeros and assigned singularities. Thus if a function have a + mat as its zeros and
c+m'<o as its singularities, where m and m' have all integral values from — oo to +00,
the simplest form is obtained by taking a constant multiple of

TTZ 7T«

tan tan —

TTZ , TTC
tan tan —

Ex. Construct a function, simply-periodic in w, having zeros given by (m+^)o> and

)o> and singularities by (m + i)co and (m + §) co.
The irreducible zeros are ^co and f w ; the irreducible singularities are \u> and §«. Now

f.TTZ \ ( TTZ , \

I tan tan ATT I I tan tan |TT I

/ \ <" / \ M /

7TZ \ ( TTZ „ \

tan tan JTT ] ( tan tan |TT I

/ \ /

is evidently a function, initially satisfying the required conditions. But, as tari^r is
infinite, we divide out by it and absorb it into A' as a factor ; the function then takes
the form

1 + tan -

3-tan'7^

60

We shall not consider simply-periodic functions, which have essential
singularities elsewhere than at z = <x> ; adequate investigation will be found
in the second part of Guichard's memoir, (I.e., p. 147). But before leaving the
consideration of the present class of functions, one remark may be made. It
was proved, in our earlier investigations, that uniform functions can be
expressed as infinite series of functions of the variable and also as infinite
products of functions of the variable. This general result is true when the
functions in the series and in the products are simply-periodic in the same
period. But the function, so represented, though periodic in that common
period, may also have another period : and, in fact, many doubly-periodic
functions of different kinds (§ 136) are often conveniently expressed as infinite
converging series or infinite converging products of simply-periodic functions.

Any detailed illustration of this remark belongs to the theory of elliptic functions : one
simple example must suffice.

, ima'

Let the real part of - - be negative, and let q denote e " ; then the function

being an infinite converging series of powers of the simply-periodic function e " , is finite
everywhere in the plane. Evidently 6 (z) is periodic in o>, so that

= 6 (z).

218 DOUBLY-PERIODIC [114.

°

Again, 0(s + «»') = 2

the change in the summation so as to give $ (z) being permissible because the extreme
terms for the infinite values of n can be neglected on account of the assumption with
regard to q. There is thus a pseudo-periodicity for 6(z) in a period <•>'.

Similarly, if 0s(z)= q* e

2J7TZ

63(z + <a') = -e " 6(z).

Then 63(z) -r-d(z) is doubly-periodic in w and 2co', though constructed only from
functions simply-periodic in w : it is a function with an infinite number of irreducible
accidental singularities in a band.

115. We now pass to doubly-periodic functions of a single variable, the
periodicity being additive. The properties, characteristic of this important
class of functions, will be given in the form either of new theorems or
appropriate modifications of theorems, already established ; and the develop
ment adopted will follow, in a general manner, the theory given by Liouville*.
It will be assumed that the functions are uniform, unless multiformity be
explicitly stated, and that all the singularities in the finite part of the plane
are accidental "f*.

The geometrical representation of double-periodicity, explained in § 105,
will be used concurrently with the analysis; and the parallelogram of
periods, to which the variable argument of the function is referred, is a
fundamental parallelogram (§ 109) with periods J 2co and 2&>'. An angular
point £0 for the parallelogram of reference can be chosen so that neither a
zero nor a pole of the function lies on the perimeter; for the number
of zeros and the number of poles in any finite area must be finite,
otherwise they would form a continuous line or a continuous area, or thej
would be in the vicinity of an essential singularity. This choice will, ir

* In his lectures of 1847, edited by Borchardt and published in Crelle, t. Ixxxviii, (1880), pp.
277 — 310. They are the basis of the researches of Briot and Bouquet, the most complet
exposition of which will be found in their Theorie des fonctions elliptiques, (2nd ed.), pp.
239—280.

t For doubly-periodic functions, which have essential singularities, reference should be made
to Guichard's memoir, (the introductory remarks aud the third part), already quoted on p. 147, note.

J The factor 2 is introduced merely for the sake of convenience.

115.] FUNCTIONS 219

general, be made ; but, in particular cases, it is convenient to have the origin
as an angular point of the parallelogram and then it not infrequently occurs
that a zero or a pole lies on a side or at a corner. If such a point lie on a side,
the homologous point on the opposite side is assigned to the parallelogram
which has that opposite side as homologous; and if it be at an angular point,
the remaining angular points are assigned to the parallelograms which have
them as homologous corners.

The parallelogram of reference will therefore, in general, have z0, z0 + 2&>,
z0 + 2&/, z0 + 2&> + 2&>' for its angular points ; but occasionally it is desirable
to .take an equivalent parallelogram having z0 ± &> + &>' as its angular
points.

When the function is denoted by </> (2), the equations indicating the
periodicity are

<£ (z + 2<w) = (f> (z) = (f> (z + 2&/).

116. We now proceed to the fundamental propositions relating to
doubly-periodic functions.

I. Every doubly -periodic function must have zeros and infinities within
the fundamental parallelogram.

For the function, not being a constant, has zeros somewhere in the plane
and it has infinities somewhere in the plane ; and, being doubly-periodic, it
experiences within the parallelogram all the variations that it can have over
the plane.

COROLLARY. The function cannot be a rational integral function of z.

For within a parallelogram of finite dimensions an integral function has
no infinities and therefore cannot represent a doubly-periodic function.

An analytical form for <j) (z) can be obtained which will put its singu
larities in evidence. Let a be such a pole, of multiplicity n ; then we know
that, as the function is uniform, coefficients A can be determined so that the
function

f(* ~ (z-a)n~ (z-a)n-1~'"~(z-a)2~ z^a

is finite in the vicinity of a ; but the remaining poles of <j> (z) are singularities
of this modified function. Proceeding similarly with the other singularities
b, c,..., which are finite in number and each of which is finite in degree, we
have coefficients A, B, C,... determined so that

A^< V i? K'

9 (z) — 2, f Z T r

is finite in the vicinity of every pole of <f) (z) within the parallelogram and
therefore is finite everywhere within the parallelogram. Let its value be

220 PROPERTIES [116.

%(X); then for points lying within the parallelogram, the function <f>(z) is
expressed in the form

+ A* +

^ 1 1

A,

T T ;

2 — a (

ft

+ 1 i

\9 1 ' ' ' ' /

z - a> (
B2

z - a)n
Bm

X. ' /

£—6 (

7 \ 0 ' • • • 1 /

z-b)m

H,

_L _L

#2

S±i

z-h^ (z-h? ' r (z-h)1'

But though <£ (^) is periodic, ^ (2?) is not periodic. It has the property of
being finite everywhere within the parallelogram ; if it were periodic, it
would be finite everywhere, and therefore could have only a constant value ;
and then <f> (z) would be an algebraical meromorphic function, which is not
periodic. The sum of the fractions in $ (z) may be called the fractional
part of the function : owing to the meromorphic character of the function,
it cannot be evanescent.

The analytical expression can be put in the form
(z - a)~n (z - 6)-™. . .(z - h)~l F(z\

where F(z) is finite everywhere within the parallelogram. If a, /3, ..., ij be
all the zeros, of degrees v, p, ..., X, within the parallelogram, then

F(z) = (z-a)v(z-py ...(z-^G(z\

where G (z) has no zero within the parallelogram ; and so the function can
be expressed in the form

(z-a)n(z-b}m...(z-h)1 G^'

where G (z) has no zero and no infinity within the parallelogram or on its
boundary ; and G {z) is not periodic.

The order of a doubly-periodic function is the sum of the multiplicities
of all the poles which the function has within a fundamental parallelogram;
and, the sum being n, the function is said to be of the nth order. All
these singularities are, as already remarked, accidental; it is convenient
to speak of any particular singularity as simple, double, . . . according to its
multiplicity.

If two doubly-periodic functions u and v be such that an equation

is satisfied for constant values of A, B, C, the functions are said to be
equivalent to one another. Equivalent functions evidently have the same
accidental singularities in the same multiplicity.

II. The integral of a doubly-periodic function round the boundary of a
fundamental parallelogram is zero.

116.]

OF DOUBLY-PERIODIC FUNCTIONS

221

Let ABCD be a fundamental parallelogram, the boundary of it being
taken so as to pass through no pole of the
function. Let A be z0, B be z0+2ca, and* <=

D be z0 + 2a)': then any point in AB is / °

/Q* Q,

where £ is a real quantity lying between 0 and 1 ;
and therefore the integral along AB is

rl

Any point in EG is z0 + 2<w + 2&>'£, where £ is a real quantity lying between 0
and 1 ; therefore the integral along BC is

(o 'dt,

o

since <^> is periodic in 2&).

Any point in DC is s + 2o>' + 2<wZ, where < is a real quantity lying
between 0 and 1 ; therefore the integral along CD is

f°

J 1

2ft)'

= - I

J o

Similarly, the integral along DA is

= - I cf> Oo + 2o>'«) 2w'^.

J o

Hence the complete value of the integral, taken round the parallelogram, i

fi
= <j>(z0

Jo

which ^ is manifestly zero, since each of the integrals is the integral of
a continuous function.

COROLLARY. Let ty(z) be any uniform function of zt not necessarily
doubly-periodic, but without singularities on the boundary. Then the

* The figure implies that the argument of w' is greater than the argument of w, a
hypothesis which, though unimportant for the present proposition, must be taken account of
hereafter (e.g., § 129).

222 INTEGRAL RESIDUE [116.

integral jty (z) dz taken round the parallelogram of periods is easily seen
to be

n ri

•^ (z(} + Scot) 2udt + I ^(z0 + 2a> + 2m't) 2a>'dt

Jo J o

ri ri

- V (*o + 2o>' + 2a>t) 2(odt - ^ (z0 + 2to't) 2w'dt ;

Jo Jo

or, if we write

/• ri ri

then U- (2) ^ = I I/TJ (>0 + 2w't) 2m dt - ^ (z, + 2wt) 2(odt,

J Jo Jo

where on the left-hand side the integral is taken positively round the
boundary of the parallelogram and on the right-hand side the variable t
in the integrals is real.

The result may also be written in the form

r rD rx

\-^r(z)dz=\ ^ (z) dz — I -»K (z) dz,

J J A J A

the integrals on the right-hand side being taken along the straight lines AD
and AB respectively.

Evidently the foregoing main proposition is established, when -^ (£) and
T/r2 (f) vanish for all values of £.

III. If a doubly -periodic function $(z) have infinities Oj, a2, ... within
the parallelogram, and if Al, A2, ... be the coefficients of (z — e^)"1, (z — a^r1, . . .
respectively in the fractional part of (j> (z) when it is expanded in the parallelo
gram, then

A1 + A2+...=0.

As the function <f>(z) is uniform, the integral f(f>(z)dz is, by (§ 19, II.), the
sum of the integrals round a number of curves each including one and only
one of the infinities within that parallelogram.

Taking the expression for (f>(z) on p. 220, the integral Amf(z — a)~mdz
round the curve enclosing a is 0, if m be not unity, and is Z>jriAl, if m be
unity; the integral Kmf(z — k)~mdz round that curve is 0 for all values of m
and for all points k other than a ; and the integral /^ (z) dz round the curve
is zero, since % (z) is uniform and finite everywhere in the vicinity of a. Hence
the integral of <£ (z) round a curve enclosing c^ alone of all the infinities is

Similarly the integral round a curve enclosing a.2 alone is 27riA.2; and so
on, for each of the curves in succession.

Hence the value of the integral round the parallelogram is

2-rnZA.

116.] OF FUNCTIONS OF THE SECOND ORDER 223

But by the preceding proposition, the value of /(/> (2) dz round the parallelo
gram is zero ; and therefore

This result can be expressed in the form that the sum of the residues* of a
doubly -periodic function relative to a fundamental parallelogram of periods
is zero.

COROLLARY 1. A doubly-periodic function of the first order does not

exist.

Let such a function have a for its single simple infinity. Then an
expression for the function within the parallelogram is

A

^-a + *^>

where ^ (2) is everywhere finite in the parallelogram. By the above propo
sition, A vanishes ; and so the function has no infinity in the parallelogram.
It therefore has no infinity anywhere in the plane, and so is merely a
constant : that is, qua function of a variable, it does not exist.

COROLLARY 2. Doubly-periodic functions of the second order are of two
classes.

As the function is of the second order, the sum of the degrees of the
infinities is two. There may thus be either a single infinity of the second
degree or two simple infinities.

In the former case, the analytical expression of the function is

where a is the infinity of the second degree and ^ (z) is holomorphic within
the parallelogram. But, by the preceding proposition, A1 = 0; hence the
analytical expression for a doubly-periodic function with a single irreducible
infinity a of the second degree is

(z - of T * v

within the parallelogram. Such functions of the second order, which have
only a single irreducible infinity, may be called the first class.
In the latter case, the analytical expression of the function is

where c, and c2 are the two simple infinities and x(z} ig finite within the
parallelogram. Then

See p. 42.

224 PROPERTIES OF FUNCTIONS [116.

so that, if Cl = - C.2 = C, the analytical expression for a doubly-periodic
function with two simple irreducible infinities a1 and «2 ig

n
G

( 1 1

( -

\z-a-L z -

within the parallelogram. Such functions of the second order, which have
two irreducible infinities, may be called the second class.

COROLLARY 3. If within any parallelogram of periods a function is
only of the second order, the parallelogram is fundamental.

COROLLARY 4. A similar division of doubly -periodic functions of any
order into classes can be effected according to the variety in the constitution of
the order, the number of classes being the number of partitions of the order.

The simplest class of functions of the nth order is that in which the
functions have only a single irreducible infinity of the nth degree. Evi
dently the analytical expression of the function within the parallelogram is

G, G, Gn

(z - a)2 (z - a)3 (z - a)n * ^ ''

where ^ (z) is holomorphic within the parallelogram. Some of the coefficients
G may vanish ; but all may not vanish, for the function would then be finite
everywhere in the parallelogram.

It will however be seen, from the next succeeding propositions, that the
division into classes is of most importance for functions of the second
jrder.

IV. Two functions, which are doubly-periodic in the same periods*, and
which have the same zeros and the same infinities each in the same degrees
respectively, are in a constant ratio.

Let <f) and ^ be the functions, having the same periods; and let a of
degree v, /3 of degree fi, ... be all the irreducible zeros of <£ and T/T; arid a of
degree n, b of degree m, ... be all the irreducible infinities of <f> and of ty.
Then a function G (z), without zeros or infinities within the parallelogram,
exists such that

, , , = (z-a)v(z-py ... G _

and another function H(z), without zeros or infinities within the parallelo
gram, exists such that

Hence *(*)_<?(*)

-

Now the function on the right-hand side has no zeros in the parallelogram,
for G has no zeros and H has no infinities ; and it has no infinities in the

* Such functions will be called homoperiodic.

116.] OF THE SECOND ORDER 225

parallelogram, for G has no infinities and H has no zeros : hence it has
neither zeros nor infinities in the parallelogram. Since it is equal to the
function on the left-hand side, which is a doubly-periodic function, it has no
zeros and no infinities in the whole plane ; it is therefore a constant, say
A. Thus*

V. Two functions of the second order, doubly -periodic in the same periods
and having the same infinities, are equivalent to one another.

If one of the functions be of the first class in the second order, it has one
irreducible double infinity, say at a ; so that we have

where %(z) is finite everywhere within the parallelogram. Then the other
function also has z = a for its sole irreducible infinity and that infinity is of
the second degree ; therefore we have

TT

where ^ (z) is finite everywhere within the parallelogram. Hence

Now x and %x are finite everywhere within the parallelogram, and therefore
so is H% — Gfo. But H% — Gfo, being equal to the doubly-periodic function
H(j) — Gijr, is therefore doubly-periodic ; as it has no infinities within the
parallelogram, it consequently can have none over the plane and therefore it
is a constant, say 7. Thus

proving that the functions <j> and ty are equivalent.

If on the other hand one of the functions be of the second class in the
second order, it has two irreducible simple infinities, say at 6 and c, so that
we have

where 6(z) is finite everywhere within the parallelogram. Then the other
function also has z = b and z = c for its irreducible infinities, each of them
being simple ; therefore we have

where 6l (z) is finite everywhere within the parallelogram. Hence

(z) - Cty (z) = De (z) - Cei (z}.

* This proposition is the modified form of the proposition of § 52, when the generalising
exponential factor has been determined so as to admit of the periodicity.

F. 15

226 IRREDUCIBLE ZEROS [116.

The right-hand side, being finite everywhere in the parallelogram, and equal
to the left-hand side which is a doubly-periodic function, is finite everywhere
in the plane ; it is therefore a constant, say B, so that

proving that <£ and ty are equivalent to one another.

It thus appears that in considering doubly-periodic functions of the second
order, homoperiodic functions of the same class are equivalent to one another
if they have the same infinities ; so that, practically, it is by their infinities
that homoperiodic functions of the second order and the same class are dis
criminated.

COROLLARY 1. If two equivalent functions of tlie second order have one
zero the same, all their zeros are the same.

For in the one class the constant /, and in the other class the constant B,
is seen to vanish on substituting for z the common zero ; and then the two
functions always vanish together.

COROLLARY 2. If two functions, doubly-periodic in the same periods but
not necessarily of the second order, have the same infinities occurring in such a, j
way that the fractional parts of the two functions are the same except as to a
constant factor, the functions are equivalent to one another. And if, in
addition, they have one zero common, then all their zeros are common, so
that the functions are then in a constant ratio.

COROLLARY 3. If two functions of the second order, doubly-periodic in(
the same periods, have their zeros the same, and one infinity common, they are ^
in a constant ratio.

VI. Every doubly -periodic function has as many irreducible zeros as it
has irreducible infinities.

Let <£ (z) be such a function. Then

z +h — z

is a doubly-periodic function for any value of h, for the numerator is doubly-
periodic and the denominator does not involve z ; so that, in the limit when
h = 0, the function is doubly-periodic, that is, </>' (z) is doubly-periodic.

Now suppose <f>(z) has irreducible zeros of degree m1 at a1} ra2 at a2, ...,
and has irreducible infinities of degree /^ at «1} yu,2 at «2, ... ; so that the
number of irreducible zeros is Wj + ra2 + . . . , and the number of irreducible
infinities is ^1 + /i2 + ..., both of these numbers being finite. It has been
shewn that <£ {z) can be expressed in the form

116.] AND IRREDUCIBLE INFINITIES 227

whore F(z) has neither a zero nor an infinity within, or on the boundary of,
the parallelogram of reference.

Since F(z) has a value, which is finite, continuous and different from zero

Tjlt / \

everywhere within the parallelogram or on its boundary, the function -p4-r

* W
is not infinite within the same limits. Hence we have

rr - ~ — ...

9 (z) z—a± z — «2

+ -* + =*. + ..

z — ttj z — a2

where g (z) has no infinities within, or on the boundary of, the parallelogram
of reference. But, because <f> (z) and <f> (z) are doubly-periodic, their quotient
is also doubly-periodic ; and therefore, applying Prop. II., we have

m^ + w2 + . . . — ^ — p2 — . . . = 0,
that is, m1+m2 + ... = fj,! + fi2+ ...,

or the number of irreducible zeros is equal to the number of irreducible
infinities.

COROLLARY I. The number of irreducible points for which a doubly -
periodic function assumes a given value is equal to the number of irreducible
zeros.

For if the value be A, every infinity of $(z) is an infinity of the doubly-
periodic function $ (z) — A ; hence the number of the irreducible zeros of the
latter is equal to the number of its irreducible infinities, which is the same as
the number for <£ (z} and therefore the same as the number of irreducible
zeros of <£ (z). And every irreducible zero of <£ (z} — A is an irreducible
point, for which <£ (z) assumes the value A.

COROLLARY II. A doubly-periodic function with only a single zero does
not exist; a doubly -periodic function of the second order has two zeros; and,
generally, the order of a function can be measured by its number of irreducible
zeros.

Note. It may here be remarked that the doubly-periodic functions
(§ 115), that have only accidental singularities in the finite part of the
plane, have z = oo for an essential singularity. It is evident that for infinite
values of z, the finite magnitude of the parallelogram of periods is not
recognisable ; and thus for z = GO the function can have any value, shewing
that z = oo is an essential singularity.

VII. Let a1} a2)... be the irreducible zeros of a function of degrees
w1; m2, ... respectively ; a1} «2, ... its irreducible infinities of degrees /^, /u,2, ...
respectively; and z1,z2,... the irreducible points where it assumes a value c,
which is neither zero nor infinity, their degrees being M1} M.2) ... respectively.

15—2

228 IRREDUCIBLE ZEROS [116.

Then, except possibly as to additive multiples of Hie periods, the quantities
2 mrar, 2 UrCir and 2 Mrzr are equal to one another, so that

r=l r=l r=l

2 mrar = 2 Mrzr = 2 prctr (mod. 2o>, 2&/)-

r=l r=l r=l

Let (/> (/) be the function. Then the quantities which occur are the sums
of the zeros, the assigned values, and the infinities, the degree of each being
taken account of when there is multiple occurrence ; and by the last
proposition these degrees satisfy the relations

The function <f)(z) — c is doubly-periodic in 2«u and 2&>' ; its zeros are
z1} z.2, ... of degrees M1} M^,... respectively; and its infinities are ctl, «2, ... of
degrees /i1} yn2, •••, being the same as those of <£(Y). Hence there exists a
function G(z), without either a zero or an infinity lying in the parallelogram
or on its boundary, such that </> 0) - c can be expressed in the form

^l*1C.(*I*a>!'" G (*)

for all points not outside the parallelogram ; and therefore, for points in that

region

<f>'(Y) ^ Mr ^ *r G'(z)

\ /~** / \ •

/ \

<j)(z) — C r=l Z — Zr Z— O.r

Hence

z$(z) ~ Mrz v prz zG' (z)

. . >. - — 2< - --- 2* --- 1 — .~ , .
$(z) — C r=l z — zr Z— ar W (*)

= 2 Jfr+ 2

,

~r

* ~r /-v / -. ,

=\Z— Zr Z—OLr (r(z)

2 Mr= 2 nr.

r=l r=l

Integrate both sides round the boundary of the fundamental parallelogram.
Because G (z) has no zero and no infinity in the included region and does not
vanish along the curve, the integral

'zG'(z)

I

dz

G(z)

vanishes. But the points z{ and 04 are enclosed in the area ; and therefore
the value of the right-hand side is

2iri 2 Mrzr — Ziri 2 /V*r,

so

that

\Z) — c
the integral being extended round the parallelogram.

116.] AND IRREDUCIBLE INFINITIES 229

zd>' (z)

Denoting the subject of integration , by/(^), we have

<p(z) — c

-/«=*" -

and therefore, by the Corollary to Prop. II., the value of the foregoing
integral is

*• r £¥-*-** r £¥-*•

JA<f>(Z)-C JA(j)(z)-G

the integrals being taken along the straight lines AD and AB respectively
(fig. 33, p. 221).

Let w — <f)(z) — c; then, as z describes a path, w will also describe a single
path as it is a uniform function of z. When z moves from A to D, w moves
from (j>(A)-c by some path to (f>(D) — c, that is, it returns to its initial
position since <f> (D) = <f> (A) ; hence, as z describes AD, w describes a simple
closed path, the area included by which may or may not contain zeros and
infinities of w. Now

dw = <f>' (z) dz,

CD <£' (z\

and therefore the integral I ,,\ dz is equal to
* JAJ>(*)-C

I

dw
w

taken in some direction round the corresponding closed path for w. This
integral vanishes, if no w-zero or w-infinity be included within the area
bounded by the path ; it is + Im'iri, if m be the excess of the number of
included zeros over the number of included infinities, the + or — sign being
taken with a positive or a negative description ; hence we have

where m is some positive or negative integer and may be zero. Similarly

where n is some positive or negative integer and may be zero.

Thus 27Ti (2,MrZr ~ 2/V*,-) = 2w . 2w7n — 2a)' . Smri,

and therefore ^Mrzr — ^prir = 2ma> — 2?io>'

= 0 (mod. 2&), 2o>').

Finally, since ^Mrzr = 2/v*r whatever be the value of c, for the right-hand

230 DOUBLY-PERIODIC FUNCTIONS [116.

side is independent of c, we may assign to c any value we please. Let the
value zero be assigned ; then ^Mrzr becomes Smrar, so that

^mrar = "2/j,rf*r (mod. 2&>, 2&/).

The combination of these results leads to the required theorem*, expressed
by the congruences

2 mrar = 2 Mrzr = 2 ^r^r (mod. 2o>, 2&>').

r=l r=l r=l

Note. Any point within the parallelogram can be represented in the
form z0 + a2&> + 62&>', where a and 6 are real positive quantities less than

unity. Hence

2 Mrzr = Az'2a> + Bz2a>/ + z£Mr,

where J. and B are real positive quantities each less than 27lfr, that is, less
than the order of the function.

In particular, for functions of the second order, we have

z1 + z, = Az 2&> + Bz 2&/ + 2.2-0,
where Az and Bz are positive quantities each less than 2. Similarly, if a and

b be the zeros,

a + b = Aa 2w + £a 2w' + 2*o,

where J.ffl and Ba are each less than 2 ; hence, if

^i + ^2 — a — b — w2w + m'2o>',

then w may have any one of the three values - 1, 0, 1 and so may m', the
simultaneous values not being necessarily the same.

Let a and ft be the infinities of a function of the second class ; then
a + /3 — a — b = ?i2&) + n"2w',

where n and ri may each have any one of the three values — 1, 0, 1. By
changing the origin of the fundamental parallelogram, so as to obtain a
different set of irreducible points, we can secure that n and n' are zero,

and then

a + @ = a+b.

Thus, if n be 1 with an initial parallelogram, so that

a + /3 = a + &+2&>,

we should take either /3 - 2&> = {¥, or a - 2&> = a', according to the position of
a and /3, and then have a new parallelogram such that
a + @' = a + b, or a' + ft = a + b.

The case of exception is when the function is of the first class and has a
repeated zero.

* The foregoing proof is suggested by Konigsberger, Theorie der elliptischen Functionen,
t. i, p. 342 ; other proofs are given by Briot and Bouquet and by Liouville, to whom the adopted
form of the theorem is due. The theorem is substantially contained in one of Abel's general
theorems in the comparison of transcendents.

116.] OF THE SECOND ORDER 231

VIII. Let $ (z) be a doubly -periodic function of the second order. If 7
be the one double infinity when the function is of the first class, and if a and ft
be the two simple infinities when the function is of the second class, then in the
former case

and in the latter case </> (z) — <£ (a + (3 — z).

Since the function is of the second order, so that it has two irreducible
infinities, there are two (and only two) irreducible points in a fundamental
parallelogram at which the function can assume any the same value : let
them be z and z'.

Then, for the first class of functions, we have
z + z' = 27

= 27 + 2mo> + 2wa>',

where m and n are integers ; and then, since <f)(z) = <j> (z'} by definition of z
and /, we have

<£ (z) = <£ (27 - z + 2ma)

= 0(27-4
For the second class of functions, we have

z + z = a. + /3

so that, as before,

(/> (z) = </> (a + /3 - z + 2ma) + 2wa>')

117. Among the functions which have the same periodicity as a given
function </> (z), the one which is most closely related to it is its derivative
<£' (z). We proceed to find the zeros and the infinities of the derivative of a
function, in particular, of a function of the second order.

Since (f> (z) is uniform, an irreducible infinity of degree n for </> (z) is an
irreducible infinity of degree n -f 1 for §' (z). Moreover <£' (z), being uniform,
has no infinity which is not an infinity of </> (z) ; thus the order of <£' (z) is
2(?i + l) or its order is greater than that of cj>(z) by an integer which
represents the number of distinct irreducible infinities of <£ (z), no account
being taken of their degree. If, then, a function be of order m, the order of
its derivative is not less than m + 1 and is not greater than 2m.

Functions of the second order either possess one double infinity so that
within the parallelogram they take the form

—
and then <j>' (z) = — - — + %' (*),

232 ZEROS OF THE DERIVATIVE [117.

that is, the infinity of (f>(z) is the single infinity of tf>' ' (z) and it is of the
third degree, so that cf>' (z) is of the third order ; or they possess two simple
infinities, so that within the parallelogram they take the form

and then f W = - G - - _ + x' (,),

that is, each of the simple infinities of <£ (z) is an infinity for </>' (z) of the
second degree, so that <£' (z) is of the fourth order.

It is of importance (as will be seen presently) to know the zeros of
the derivative of a function of the second order.

For a function of the first class, let 7 be the irreducible infinity of the
second degree ; then we have

and therefore $'(2) = — </>' (^7 — z).

Now </>' (z) is of the third order, having 7 for its irreducible infinity in the
third degree : hence it has three irreducible zeros.

In the foregoing equation, take z = 7 : then

</>' (7) = -$' (7),

shewing that 7 is either a zero or an infinity. It is known to be the only
infinity of <£' (z).

Next, take z = 7 + &> ; then

<£' (7 + &)) = — $' (7 — a>)

= - <£' (7 + G>),

shewing that 7 + &> is either a zero or an infinity. It is known not to be an
infinity ; hence it is a zero.

Similarly 7 + &/ and 7 + <u + &/ are zeros. Thus three zeros are obtained,
distinct from one another ; and only three zeros are required ; if they be not
within the parallelogram, we take the irreducible points homologous with
them. Hence :

IX. The three zeros of the derivative of a function, doubly -periodic in
2eo and 2eo' and having 7 for its double (and only) irreducible infinity, are

7 + &), 7 + eo', 7 + w + w .

For a function of the second class, let a and /3 be the two simple
irreducible infinities; then we have

and therefore <f>' (z)= — <f>' (a + ft — z).

117.] OF A DOUBLY-PERIODIC FUNCTION 233

Now (j) (z) is of the fourth order, having a and ft as its irreducible
infinities each in the second degree ; hence it must have four irreducible
zeros.

In the foregoing equation, take z = \(VL + ft) ; then

shewing that | (a + /3) is either a zero or an infinity. It is known not to be
an infinity ; hence it is a zero.

Next, take z = £ (a + (3) + w ; then

f(}(«t£)+«} --+'{*(«+£)-••]

= - <£' & (a + £) - to + 2&>j

—.+'{*<«+£)+••},

shewing that |(a + /3) + &> is either a zero or an infinity. As before, it is
a zero.

Similarly i (a + /3) + &>' and i (a + /3) -f &> + &>' are zeros. Four zeros are
thus obtained, distinct from one another; and only four zeros are required.
Hence :

X. The four zeros of the derivative of a function, doubly-periodic in 2&>
and 2o)' and having a and /3 for its simple (and only) irreducible infinities, are

i(a + /3), i(a + /3) + a>, i(a + /3) + ft>', |- (a + /3) + w + a/.

The verification in each of these two cases of Prop. VII., that the sum of
the zeros of the doubly-periodic function <£' (z) is congruent with the sum of
its infinities, is immediate.

Lastly, it may be noted that, if zl and z^ be the two irreducible points for
which a doubly -periodic function of the second order assumes a given value,
then the values of its derivative for z1 and for z% are equal and opposite. For

(j> (z) = <f> (a + /3 - z) = cf> (z, + z.2 - z),
since zl + z., = a + (3 ; and therefore

<f> (z) = -$' (z, + z.2- z),
that is, <£' (zl) = — </>' (z2),

which proves the statement.

118. We now come to a different class of theorems.

XI. Any doubly -periodic function of the second order can be expressed
algebraically in terms of an assigned doubly-periodic function of the second
order, if the periods be the same.

The theorem will be sufficiently illustrated and the line of proof
sufficiently indicated, if we express a function (/> (z) of the second class, with
irreducible infinities a, ft and irreducible zeros a, b such that a + (3 = a + b, in

234 FUNCTIONS [118.

terms of a function <£ of the first class with 7 as its irreducible double
infinity.

n .. , ,.

Consider a function

Q (z + h) _

A zero of <X> (z + h) is neither a zero nor an infinity of this function ; nor
is an infinity of <1> (z + h) a zero or an infinity of the function. It will have
a and 6 for its irreducible zeros, if

a + h = h',
b + h + h' = 27 ;

and these will be the only zeros, for <E> is of the second order. It will have o
and yS for its irreducible infinities, if

and these will be the only infinities, for <£ is of the second order. These
equations are satisfied by

Hence the assigned function, with these values of h, has the same zeros
and the same infinities as $>(z); and it is doubly-periodic in the same periods.
The ratio of the two functions is therefore a constant, by Prop. IV., so that

c|> (z + h) — <I> (h')

If the expression be required in terms of <& (z) alone and constants, then
<j> (z 4. h} must be expressed in terms of <I> (z) and constants which are values
of <X> (z) for special values of z. This will be effected later.

The preceding proposition is a special case of a more general theorem
which will be considered later ; the following is another special case of that
theorem : viz. :

XII. A doubly -periodic function with any number of simple infinities can
be expressed either as a sum or as a product, of functions of the second order
and the second class which are doubly-periodic in the same periods.

Let «j, «2, ..., an be the irreducible infinities of the function <£, and
suppose that the fractional part of <t> (z) is

•A-i , A2 [ ^ i-+ ^n

z — ttj z — «2 z — an '

with the condition A1 + A2 + + An = Q. Let <j>n(z) be a function,

doubly-periodic in the same periods, with a,-, a,- as its only irreducible infinities,

118.] OF THE SECOND ORDER 235

supposed simple; where i and j have the values 1, ,n. Then the

fractional parts of the functions ^>j, (z), <£23 (z), . . . are

0,

G,

i z — a.,

I
\z — «2 ^ — «,

respectively; and therefore the fractional part of

^!^ / \ ,
• 0» W +

is Al An- An~l

z-a.! Z-CL, z-cin-T. z-an

•Ai An_^ An

= - -+...+- - + — ^,

Z-Cl! Z- «„_! Z - Ctn

n

since S -4* = 0. This is the same as the fractional part of <l> (z); and therefore

- <^>23 (f) - ... - -~

has no fractional part. It thus has no infinity within the parallelogram ; it
is a doubly-periodic function and therefore has no infinity anywhere in the
plane; and it is therefore merely a constant, say B. Hence, changing the
constants, we have

$>(z)-B^(z}-B.><t>v(z)-...-Bn-,<t>n-,,n(z} = B,

giving an expression for <$> (z} as a linear combination of functions of the
second order and the second class. But as the assignment of the infinities is
arbitrary, the expression is not unique.

For the expression in the form of a product, we may denote the n
irreducible zeros, supposed simple, by «!,...,«„. We determine n - 2 new
irreducible quantities c, such that

C2=

Cn—2 — &n—\ ~r Cn—3 ~ Q"n—i >
Cln = ttn + C_ — Q"—

n

this being possible because 2 o^ = 2 ar ; and we denote by $ (z ; a, ft ; e, f) a

»•=! r=l

function of .gr, which is doubly-periodic in the periods of the given function,

ALGEBRAICAL RELATIONS [118.

has a and $ for simple irreducible infinities and has e and / for simple
irreducible zeros. Then the function

<f)(z; al5 «2 ; «i, Ci) <f> (z ; as, ci ; 0-2, c2) ...<£ (2 ; «n, cn_2 ; an_l5 an)
has neither a zero nor an infinity at c1} at c2, ..., and at cn_2 ; it has simple
infinities at al} a2, ..., an, and simple zeros at alt a2, ..., an-1} an. Hence it
has the same irreducible infinities and the same irreducible zeros in the same
degree as the given function <£ (z) ; and therefore, by Prop. IV., <I> (z) is
a mere constant multiple of the foregoing product.

The theorem is thus completely proved.

Other developments for functions, the infinities of which are not simple,
are possible ; but they are relatively unimportant in view of a theorem,
Prop. XV., about to be proved, which expresses any periodic function in
terms of a single function of the second order and its derivative.

XIII. If two doubly -periodic functions have the same periods, they are
connected by an algebraical equation.

Let u be one of the functions, having n irreducible infinities, and v be
the other, having m irreducible infinities.

By Prop. VI., Corollary I., there are n irreducible values of z for a value
of u; and to each irreducible value of z there is a doubly-infinite series of.
values of z over the plane. The function v has the same value for all the
points in any one series, so that a single value of v can be associated uniquely
with each of the irreducible values of z, that is, there are n values of v for
each value of u. Hence, (§ 99), v is a root of an algebraical equation of the
nth degree, the coefficients of which are functions of u.

Similarly u is a root of an algebraical equation of the mth degree, the
coefficients of which are functions of v.

Hence, combining these results, we have an algebraical equation between
u and v of the nth degree in v and the mth in u, where m and n are the
respective orders of v and u.

COROLLARY I. If both the functions be even functions of z, then n and m
are even integers ; and the algebraical relation between u and v is of degree ^n
in v and of degree ^m in u.

COROLLARY II. If a function u be doubly-periodic in &> and &>', and a
function v be doubly -periodic in fl and U', where

n = mca + nta, I!' = m'w + nw! ,
m, n, m', n being integers, then there is an algebraic relation between u and v.

119. It has been proved that, if a doubly-periodic function u be of order m,
then its derivative du/dz is doubly-periodic in the same periods and is of an
order n, which is not less than m + 1 and not greater than 2?/i. Hence, by

119.] BETWEEN HOMOPERIODIC FUNCTIONS 237

Prop. XIII., there subsists between u and u an algebraical equation of order m
in u' and of order n in u; let it be arranged in powers of u' so that it takes
the form

U" u'm _j_ JJ u'm—i _i _ _ i U _ u'2 i JJ _u' i JJ __ Q

where U0, U1} ... , Um are rational integral algebraical functions of u one at
least of which must be of degree n.

Because the only distinct infinities of u' are infinities of u, it is impossible
that u' should become infinite for finite values of u: hence U0 = 0 can have no
finite roots for u, that is, it is a constant and so it may be taken as unity.

And because the m values of z, for which u assumes a given value, have
their sum constant save as to integral multiples of the periods, we have

corresponding to a variation 8u ; or

du du du

f/7/

Now — is one of the values of u' corresponding to the value of u, and so for

the others ; hence

3 1

r=i ur
that is, by the foregoing equation,

" m— i

= 0,

un.

and therefore Um-^ vanishes. Hence :

XIV. There is a relation, between a doubly -periodic function u of order m
and its derivative, of the form

u'm + U^'™-1 + ...+ U^u'* + Um = 0,

where Ul}..., Um_2, Um are rational integral algebraical functions of u, at
least one of which must be of degree n, the order of the derivative, and n is
not less than m + 1 and not greater than 2m.

Further, by taking v = - , which is a function of order m because it has the

Uj

m irreducible zeros of u for its infinities, and substituting, we have

vf™ _ 03 U^'m~l + v*U«v'm~* - . . . ± v2"1-4 Um_2v''2 + v2"1 Um = 0.
The coefficients of this equation must be integral functions of v ; hence the
degree of Ur in u cannot be greater than 2r.

COROLLARY. The foregoing equation becomes very simple in the case of
doubly-periodic functions of the second order.

Then m = 2.

238 DIFFERENTIAL EQUATION [119.

If the function have one infinity of the second degree, its derivative has
that infinity in the third degree, and is of the third order, so that n = 3 ; and
the equation is

/y7?/\2

( ^ ) = \u? + 3/iw2 + Svu + p,
\d*J

where X, /*, v, p are constants. If 6 be the infinity, so that

A
*.£(,)_-_— + £(*),

where % (^) is everywhere finite in the parallelogram, then - = ±A ; and the

/77/

zeros of -j- are 6 + o>, 0 + &/, 6 + o> + CD' ; so that
diz

a,')} {

This is £/ie general differential equation of Weierstrasss elliptic functions.

If the function have two simple infinities a and @, its derivative has each
of them as an infinity of the second degree, and is of the fourth order, so that
n = 4 ; and the equation is

(du\* _
(dz) =

dM + c2w + >c3u + c4,
where c0, c1} c2, c3, c4 are constants. Moreover

where ^ (^) is finite everywhere in the parallelogram. Then cu = G~2 ; and

^/'i/

the zeros of -y- are ^ (a + /3), -|- (a + (3) + w, ^ (a -f /3) + cof, % (a + ft) + w + &>',
ft/2

so that the equation is

(« + 13)+ « + «}].
This is the general differential equation of Jacobis elliptic functions.

The canonical forms of both of these equations will be obtained in Chapter
XI., where some properties of the functions are investigated as special illustra
tions of the general theorems.

Note. All the derivatives of a doubly-periodic function are doubly-
periodic in the same periods, and have the same infinities as the function but
in different degrees. In the case of a function of the second order, which
must satisfy one or other of the two foregoing equations, it is easy to see that
a derivative of even rank is a rational, integral, algebraical function of u, and
that a derivative of odd rank is the product of a rational, integral, algebraical
function of u by the first derivative of u.

119.] OF DOUBLY-PERIODIC FUNCTIONS 239

It may be remarked that the form of these equations confirms the result
at the end of § 117, by giving two values of u' for one value of u, the two
values being equal and opposite.

Ex. If u be a doubly-periodic function having a single irreducible infinity of the third
degree so as to be expressible in the form

2 6
— -o + -5 + integral function of z

z z

within the parallelogram of periods, then the differential equation of the first order which
determines u is

where £74 is a quartic function of u and where a is a constant which does not vanish with 6.

(Math. Trip., Part II, 1889.)

XV. Every doubly -periodic function can be expressed rationally in terms
of a function of the second order, doubly-periodic in the same periods, and its
derivative.

Let u be a function of the second order and the second class, having the
same two periods as v, a function of the rath order ; then, by Prop. XIII.,
there is an algebraical relation between u and v which, being of the second
degree in v and the mth degree in u, may be taken in the form

Lv* - 2Mv + P = 0,

where the quantities L, M, P are rational, integral, algebraical functions of u
and at least one of them is of degree m. Taking

Lv-M=w,

we have w2 = M'2 — LP,

a rational, integral, algebraical function of u of degree not higher than 2w.

Thus w cannot be infinite for any finite value of u : an infinite value of u
makes w infinite, of finite multiplicity. To each value of u there correspond
two values of w equal to one another but opposite in sign.

Moreover w, being equal to Lv - M , is a uniform function of z, say F(z\
while it is a two-valued function of u. A value of u gives two distinct
values of z, say zl and £2 ; hence the values of w, which arise from an assigned
value of u, are values of w arising as uniform functions of the two distinct
values of z. Hence as the two values of w are equal in magnitude and
opposite in sign, we have

r(4)+J*(4)-Oi

that is, since ^ + z.2 = a. + ft where a and /3 are the irreducible infinities of u,

so that l(a + £), £(a + /3) + a>, £(a + £) + «', and £ (a + /3) + a> + a>' are either
zeros or infinities of w. They are known not to be infinities of u, and w is
infinite only for infinite values of u ; hence the four points are zeros of w.

240 RELATIONS BETWEEN [119.

But these are all the irreducible zeros of u' ; hence the zeros of u' are
included among the zeros of w.

Now consider the function w/u'. The numerator has two values equal
and opposite for an assigned value of u ; so also has the denominator. Hence
w/u' is a uniform function of u.

This uniform function of u may become infinite for
(i) infinities of the numerator,
(ii) zeros of the denominator.

But, so far as concerns (ii), we know that the four irreducible zeros of the
denominator are all simple zeros of u' and each of them is a zero of w .; hence
w/u' does not become infinite for any of the points in (ii). And, so far as
concerns (i), we know that all of them are infinities of u. Hence w/u, a
uniform function of u, can become infinite only for an infinite value of u, and
its multiplicity for such a value is finite; hence it is a rational, integral,
algebraical function of u, say N, so that

w = Nu'.

Moreover, because w2 is of degree in u not higher than 2m, and u'2 is of
the fourth degree in u, it follows that N is of degree not higher than m — 2.

We thus have Lv — M — Nu,

M+Nu M N ,

v= ~r = L + LU>

where L, M, N are rational, integral, algebraical functions of u ; the degrees
of L and M are not higher than m, and that of N is not higher than m — 2.

Note 1. The function u, which has been considered in the preceding
proof, is of the second order and the second class. If a function u of the
second order and the first class, having a double irreducible infinity, be
chosen, the course of proof is similar ; the function w has the three irreducible
zeros of u' among its zeros and the result, as before, is

w = Nu'.

But, now, w"- is of degree in u not higher than 2m and u'2 is of the third
degree in u ; hence N is of degree not higher than m — 2 and the degree of w2
in u cannot be higher than 2m — 1.

Hence, if L, M, P be all of degree m, the terms of degree 2m in LP — M2
disappear. If all of them be not of degree m, the degree of M must not be
higher than m — l ; the degree of either L or P must be m, but the degree
of the other must not be greater than m—l, for otherwise the algebraical
equation between u and v would not be of degree m in u.

We thus have

Lv2 - 2Mv + P = (), Lv - M = Nu',

119.] HOMOPERIODIC FUNCTIONS 241

where the degree of N in u is not higher than m — 2. If the degree of L be
less than TO, the degree of M is not higher than TO — 1 and the degree of P is
TO. If the degree of L be m, the degree of M may also be m provided that the
degree of P be TO and that the highest terms be such that the coefficient
of u2m in LP - M'2 vanishes.

Note 2. The theorem expresses a function v rationally in terms of u and
u : but u' is an irrational function of u, so that v is not expressed rationally
in terms of u alone.

But, in Propositions XI. and XII., it was indicated that a function such as
v could be rationally expressed in terms of a doubly-periodic function, such as
u. The apparent contradiction is explained by the fact that, in the earlier
propositions, the arguments of the function u in the rational expression and
of the function v are not the same ; whereas, in the later proposition whereby
v is expressed in general irrationally in terms of u, the arguments are the
same. The transition from the first (which is the less useful form) to the
second is made by expressing the functions of those different arguments in
terms of functions of the same argument when (as will appear subsequently, in
§ 121, in proving the so-called addition-theorem) the irrational function of u,
represented by the derivative u, is introduced.

COROLLARY I. Let H denote the sum of the irreducible infinities or of
the irreducible zeros of the function u of the second order, so that H = 2y for
functions of the first class, and O = a + /3 for functions of the second class.
Let u be represented by <f> (z) and v by ty (z), when the argument must be put
in evidence. Then

so that W-Z) =

J-j i_j ±j

Hence ^ (z) + ^ (fl - z) = 2 ^= 2R,

JL

First, if y (z) = ,Jr (ft - z\ then S = 0 and ^ (z) = R : that is, a function ^ (z),
which satisfies the equation

can be expressed as a rational algebraical meromorphic function of <f> (z) of the
second order, doubly -periodic in the same periods and having the sum of its
irreducible infinities congruent with O.

Second, if ^ (e) = - y, (fl _ z\ then R = 0 and ^ (*) = flf (*) ; that is,
function ^ (z), which satisfies the equation

16

a

242 HOMOPERIODIC FUNCTIONS [119.

can be expressed as a rational algebraical meromorphic function of <£ (z),
multiplied by 0' (z), where $ (z} is doubly-periodic in the same periods, is of the
second order, and has the sum of its irreducible infinities congruent with Q.

Third, if ty(z) have no infinities except those of u, it cannot become
infinite for finite values of u ; hence L = 0 has no roots, that is, L is a constant
which may be taken to be unity. Then i/r (z) a function of order m can be
expressed in the form

where, if the function </> (z) be of the second class, the degree of M is not
higher than m ; but, if it be of the first class, the degree of M is not higher
than m - 1 ; and in each case the degree of N is not higher than m - 2.

It will be found in practice, with functions of the first class, that these
upper limits for degrees can be considerably reduced by counting the degrees
of the infinities in

Thus, if the degree of M in u be ^ and of N be \ the highest degree of an :
infinity is either 2/t or 2X + 3 ; so that, if the order of ^ (z) be m, we should

have

m = 2/j, or m = 2\ + 3, >

according as m is even or odd.

When functions of the second class are used to represent a function ^r (z),
which has two infinities a and /3 each of degree n, then it is easy to see that
M is of degree n and N of degree n - 2 ; and so for other cases.

COROLLARY II. Any doubly -periodic function can be expressed rationally
in terms of any other function u of any order n, doubly-periodic in the same
periods, and of its derivative ; and this rational expression can always be taken
in the form

U0 + U,U' + t/X3 + • • • + Un-,u'n~\

where U0, ... , £7n-i are algebraical, rational, meromorphic functions of u.

COROLLARY III. If <f) be a doubly-periodic function, then <f> (u + v) can be
expressed in the form

where ^ is a doubly -periodic function in the same periods and of the second
order : each of the functions A, D, E is a symmetric function of^(u) and i/r (v),
and B is the same function of^(v) and ty(u) as C is of ty (u) and ty (v).

The degrees of A and E are not greater than m in ty (u) and than m in ^ (v),
where m is the order of </> ; the degree of D is not greater than m - 2 in ^ (u)
and than m - 2 in ^ (v) ; the degree of B is not greater than m - 2 in ^ (u)
and than m in ^ (v), and the degree of C is not greater than m - 2 in -^ (v)
and than m in -^ (u).

CHAPTER XI.

DOUBLY-PERIODIC FUNCTIONS OF THE SECOND ORDER.

THE present chapter will be devoted, in illustration of the preceding
theorems, to the establishment of some of the fundamental formulae relating
to doubly-periodic functions of the second order which, as has already (in
§ 119, Cor. to Prop. XIV.) been indicated, are substantially elliptic functions :
but for any development of their properties, recourse must be had to treatises
on elliptic functions.

It may be remarked that, in dealing with doubly-periodic functions, we
may restrict ourselves to a discussion of even functions and of odd functions.
For, if (/> (z) be any function, then £ {<j> (z} + <j>(— z}} is an even function, and
\ {(f>(z) — </>(— z}} is an odd function, both of them being doubly-periodic in
the periods of <f> (z) ; and the new functions would, in general, be of order
double that of <J>(z). We shall practically limit the discussion to even
functions and odd functions of the second order.

120. Consider a function <j>(z\ doubly-periodic in 2&> and 2w'; and let
it be an odd function of the second class, with a and ft as its irreducible
infinities, and a and b as its irreducible zeros*.

Then we have <£ (z) = (f> (a + /3 — z)

which always holds, and <f> (— z) = — </> (z)

which holds because <£ (z) is an odd function. Hence

<f> (a + /3 + z) = (/>(- *)

= -$(*)

so that a + ft is not a period ; and

-*(*),

To fix the ideas, it will be convenient to compare it with snz, for which 2w = 4^T, 2<a' =
a=iK', p=iK' + 2K, a-0, and b = 2K.

16—2

244 DOUBLY-PERIODIC FUNCTIONS [120.

whence 2 (a + /S) is a period. Since a -f /3 is not a period, we take a + /3 = a>,
or = &)', or = &> + w' ; the first two alternatives merely interchange &> and &>', so

that we have either

a + /3 = o),

or a + /3 = ft) + &>'.

And we know that, in general,

a + b = a + /3.
First, for the zeros : we have

so that </>(0) is either zero or infinite. The choice is at our disposal; for

- satisfies all the equations which have been satisfied by $(z) and an

</>(*)

infinity of either is a zero of the other. We therefore take

so that we have a = 0,

6 = to or &) + ft)'.
Next, for the infinities : we have

*(*)—$(-*)

and therefore <j> (- a) = - $ (a) = oo .

The only infinities of <£ are a and /3, so that either

— a= a,

or -CL = P.

The latter cannot hold, because it would give a + /3 = 0 whereas

or = &> + &/; hence

2a = 0,

which must be associated with a + /3 = w or with a + /3 = &> + &/.

Hence a, being a point inside the fundamental parallelogram, is either 0,
a), &)', or tw + &)'.

It cannot be 0 in any case, for that is a zero.

If a _|_ ^ = Wj then a cannot be tw, because that value would give ft = 0,
which is a zero, not an infinity. Hence either a = «', and then /3 = &/ + &>;
or a = &)' + &), and then /3 = ft)'. These are effectively one solution ; so that, if

a + /3 = &), we have

a, /3 = ft)', &>' + &))

and a, 6 = 0, &) ) '

jf a + /S = w + &>', then a cannot be CD f &)', because that value would give
{$ = 0, which is a zero, not an infinity. Hence either a = &> and then ft = &)',
or a = ft)' and then /3 = &). These again are effectively one solution ; so that,

if a + /3 = &) + &>', we have

a, £ = o), ft)'
and a, 6 = 0, «o + ft)')

120.] OF THE SECOND CLASS 245

This combination can, by a change of fundamental parallelogram, be made
the same as the former ; for, taking as new periods

2ft/ = 2a>'t 2fl = 2« + 2a>',
which give a new fundamental parallelogram, we have a + j3 = H, and

a, ft — &>', ft — ft/, that is, ft/, ft — 03' + 2<o'
so that a, /3 = ft/, O + a/]

and a, b = 0,

being the same as the former with O instead of &>. Hence it is sufficient to
retain the first solution alone : and therefore

a = to', ft = CD' + co,
a = 0, 6 = w.

Hence, by § 116, 1., we have

where F(z) is finite everywhere within the parallelogram.

Again, $ (z + a/) has z = 0 and z = &> as its irreducible infinities, and
it has 2 = 0)' and z = &> + &/ as its irreducible zeros, within the parallelogram
of (f) (z} ; hence

where ^ (2) is finite everywhere within the parallelogram. Thus

a function which is finite everywhere within the parallelogram ; since it is
doubly-periodic, it is finite everywhere in the plane and it is therefore a
constant and equal to the value at any point. Taking - i&/ as the point
(which is neither a zero nor an infinity) and remembering that </> is an odd
function, we have

* (*)*(* + «0 = - ft (*»')}' = p

k being a constant used to represent the value of - {<£ (^o/)}"2.

Also <j>(z + o)) = <f>(z + a + /3- 2&/)

= c/>0 + a + /3)=-(£ (z),
and therefore also <£ (&> — z) = <f) (z).

The irreducible zeros of <j>' (z) were obtained in § 117, X. In the
present example, those points are a>' + £ft>, &>' + ffc>, £&>, f &> ; so that, as
there, we have

£to'('W-{*(i)-4>(i*yito(*)-HW

where K is a constant. But

$ (f®) = 0 (2® - lft>) = (f) (-!«) = _(£ (1 a,) ;

246 DOUBLY-PERIODIC FUNCTIONS [120.

and 0(fw + &/) = <£(2a> + 2w' -.!&>- a/)

= <£(- 2 <o -to')

= — </>(£&> + &>');

so that • ,. . 4 I -

where J. is a new constant, evidently equal to {<£'(0)}2. Now, as we know
the periods, the irreducible zeros and the irreducible infinities of the function
</> (z), it is completely determinate save as to a constant factor. To determine
this factor we need only know the value of <$>(z) for any particular finite
value of z. Let the factor be determined by the condition

then, since <£(^ft>)<£(^G> + ft/) = T

by a preceding equation, we have

and then

ft' (*)}» - {f (0)}« [1 - {<£ (*)}2] [1 - fr {(/> (*)}']

Hence, since (/> (2) is an odd function, we have

<£ (z) = sn (//,£).

Evidently 2/xtu, 2/^ft)' = 4^T, 2^', where K and ^T' have the ordinary signifi
cations. The simplest case arises when /A = 1.

121. Before proceeding to the deduction of the properties of even
functions of z which are doubly-periodic, it is desirable to obtain the
addition-theorem for <f>, that is, the expression of <p (y + z) in terms of
functions of y alone and z alone.

When <f> (y + z) is regarded as a function of z, which is necessarily of the
second order, it is (§ 119, XV.) of the form

where M and L are of degree in <£ (z) not higher than 2 and N is independent
of z. Moreover y + z = a and y + z = ft are the irreducible simple infinities
of <j) (y + z) ; so that L, as a function of z, may be expressed in the form

and therefore

Z±_^(iHL^^^)}l
(z) -

121.] OF THE SECOND CLASS 247

where P, Q, R, S are independent of z but they may be functions of y. Now

</> (a - y) = </> (w' - y) = -

and <£ (/3 — y) = <j> (&>' + w — y} =

so that the denominator of the expression for <f> (y + 2) is

Since </> (z) is an odd function, <£' (#) is even ; hence

,A p -
~ */

and therefore $ (y + z) — $ (y — z) = - -

Differentiating with regard to z and then making z = 0, we have

so that, substituting for Q we have

Interchanging y and z and noting that </> (t/ — z) = — (f) (z — y), we have

md therefore d> C7y * Z} d>' (0} -

W+*)1><

which is the addition-theorem required.

Ex. If f(u) be a doubly-periodic function of the second order with infinities 61} i2,
and 0(tt) a doubly -periodic function of the second order with infinities alt a2 such that,
in the vicinity of «» (for i — 1, 2), we have

^ (M) = ,7~!r +Pi+& (u~ai) + ...... >

c6 — u-j

thon /M-/W = • i» W+* W-ft-ftl-

the periods being the same for both functions. Verify the theorem when the functions are
sn u and sn (u + v}. (Math. Trip. Part II., 1 891.)

Prove also that, for the function $ (u), the coefficients p± and p2 are equal. (Burnside.)

122. The preceding discussion of uneven doubly-periodic functions
having two simple irreducible infinities is a sufficient illustration of the

248 DOUBLY-PERIODIC FUNCTIONS [122.

method of procedure. That, which now follows, relates to doubly- periodic
functions with one irreducible infinity of the second degree ; and it will be
used to deduce some of the leading properties of Weierstrass's er-function
(of § 57) and of functions which arise from it.

The definition of the <r-function is

where fi = 2ma> + 2m'a)', the ratio of &>' : &> not being purely real, and the
infinite product is extended over all terms that are given by assigning to
m and to m' all positive and negative integral values from +00 to — oo ,
excepting only simultaneous zero values. It has been proved (and it is
easy to verify quite independently) 'that, when cr(z) is regarded as the
product of the primary factors

the doubly-infinite product converges uniformly and unconditionally for all
values of z in the finite part of the plane ; therefore the function which it
represents can, in the vicinity of any point c in the plane, be expanded in a
converging series of positive powers of z — c, but the series will only express
the function in the domain of c. The series, however, can be continued over
the whole plane.

It is at once evident that a- (z) is not a doubly-periodic function, for it has
no infinity in any finite part of the plane.

It is also evident that a (z) is an odd function. For a change of sign in z
in a primary factor only interchanges that factor with the one which has
equal and opposite values of m and of m', so that the product of the two factors
is unaltered. Hence the product of all the primary factors, being independent
of the nature of the infinite limits, is an even function ; when z is associated
as a factor, the function becomes uneven and it is a- (z).

The first derivative, a' (z), is therefore an even function ; and it is not
infinite for any point in the finite part of the plane.

It will appear that, though a- (z) is not periodic, it is connected with
functions that have 2o> and 2&>' for periods ; and therefore the plane will be
divided up into parallelograms. When the whole plane is divided up, as in
§ 105, into parallelograms, the adjacent sides of which are vectorial repre
sentations of 2w and 2&/, the function a-(z) has one, and only one, zero in
each parallelogram; each such zero is simple, and their aggregate is given
by z = £l. The parallelogram of reference can be chosen so that a zero
of <r (z} does not lie upon its boundary ; and, except where explicit account is

122.] OF THE FIRST CLASS 249

taken of the alternative, we shall assume that the argument of &>' is greater
than the argument of to, so that the real part* of w'/ia) is positive.

123. We now proceed to obtain other expressions for a- (z), and particu
larly, in the knowledge that it can be represented by a converging series in
the vicinity of any point, to obtain a useful expression in the form of a series,
converging in the vicinity of the origin.

Since er (z) is represented by an infinite product that converges uniformly
and unconditionally for all finite values of z, its logarithm is equal to the sum
of the logarithms of its factors, so that

where the series on the right-hand side extends to the same combinations of
m and m' as the infinite product for z, and, when it is regarded as a sum of

z z^ ( z\

functions o + i 7^2 + ^°£ ( ^ ~ r> ) » ^ne sei>ies converges uniformly and uncon-

__ -- \ 1 - ,

ditionally, except for points z = £l. This expression is valid for log a (z) over
the whole plane.

Now let these additive functions be expanded, as in § 82. In the imme
diate vicinity of the origin, we have

a series which converges uniformly and unconditionally in that vicinity.
Then the double series in the expression for log a (z} becomes

and as this new series converges uniformly and unconditionally for points in
the vicinity of z = 0, we can, as in § 82, take it in the form

oo ~r ( oo oo }

5" J 5* y O-n

— 4 — \ ^-> <5r »• ()

r=3 r (-00 -oo J

which will also, for such values of z, converge uniformly and unconditionally.
In § 56, it was proved that each of the coefficients

00 00

2 s n-*-,

— 00 - 00

for r = 3, 4,..., is finite, and has a value independent of the nature of the
infinite limits in the summation. When we make the positive infinite limit
for m numerically equal to the negative infinite limit for m, and likewise for

This quantity is often denoted by ffi ( . - J .

250 WEIERSTRASS'S [123.

ra', then each of these coefficients determined by an odd index r vanishes,
and therefore it vanishes in general. We then have

log a- 0) = log z - I* 22ft-4 - ^ 22ft-6 - ^ 22ft-8

a series which converges uniformly and unconditionally in the vicinity of the
origin.

The coefficients, which occur, involve «o and «', two independent constants.
It is convenient to introduce two other magnitudes, g.2 and g3, denned by the

equations

#2= 6022ft-4, #3 = 140220-0,

so that g2 and </3 are evidently independent of one another; then all the
remaining coefficients are functions* of g.2 and g3. We thus have

and therefore <r (z) = ze m

where the series in the index, containing only even powers of z, converges

uniformly and unconditionally in the vicinity of the origin.

It is sufficiently evident that this expression for a- (z) is an effective
representation only in the vicinity of the origin ; for points in the vicinity of
any other zero of cr (z), say c, a similar expression in powers of z - c instead
of in powers of z would be obtained.

124. From the first form of the expression for log cr (z), we have

o-(z) z _«, _

where the quantity in the bracket on the right-hand side is to be regarded as
an element of summation, being derived from the primary factor in the
product-expression for cr (z\

We write £(z) = , ^ ,

so that %(z) is, by § 122, an odd function, a result also easily derived from the
foregoing equation ; and so

This expression for £ (z) is valid over the whole plane.
Evidently £ (z) has simple infinities given by

for all values of ra and of m between + oo and - oo , including simultaneous
zeros. There is only one infinity in each parallelogram, and it is simple ; for
the function is the logarithmic derivative of a (z\ which has no infinity and

* See Quart. Journ., vol. xxii., pp. 4, 5. The magnitudes g2 and g3 are often called the
invariants.

124.] ELLIPTIC FUNCTION 251

only one zero (a simple zero) in the parallelogram. Hence %(z) is not a
doubly-periodic function.

For points, which are in the immediate vicinity of the origin, we have

but, as in the case of cr(z), this is an effective representation of %(z) only
in the vicinity of the origin ; and a different expression would be used for
points in the vicinity of any other infinity.

We again introduce a new function g> (z) defined by the equation

Because £ is an odd function, $ (z) is an even function ; and

where the quantity in the bracket is to be regarded as an element of
summation. This expression for $ (z) is valid over the whole plane.
Evidently |p (z) has infinities, each of the second degree, given by z = fl,
for all values of m and of m between -f oo and - oo , including simultaneous
zeros ; and there is one, and only one, of these infinities in each parallelogram.
One of these infinities is the origin; using the expression which represents
log a- (z) in the immediate vicinity of the origin, we have

= -2 + 20 9** + ^ 9*?+ • • •

for points z in the immediate vicinity of the origin. A corresponding
expression exists for g> (z) in the vicinity of any other infinity.

125. The importance of the function $ (z) is due to the following
theorem : —

The function $> (z) is doubly-periodic, the periods being 2<w and 2&/.
Wo have -l

where the doubly-infinite summation excludes simultaneous zero values, and
the expression is valid over the whole plane. Hence

+ ^-n - Si

252 WEIERSTRASS'S [125.

so that

obtained by combining together the elements of the summation in g> (z + 2<w)
and |p (,z). The two terms, not included in the summation, can be included,
if we remove the numerical restriction as to non-admittance of simultaneous
zero values for m and m'\ and then

2.) - f (,) = 2

_

where now the summation is for all values of m and of m' from + oo to — oo .
Let q denote the infinite limit of m, and p that of m'. Then terms in the
first fraction, for 0 = 2 (mm + m'w'}, are the same as terms in the second for
£1 = 2 (m — l)w + 2m' w ; cancelling these, we have

m'=p

-fC =

where q is infinite. But

?r)2 sin2 c '
and therefore

»»' = p = oo 1 ^.2

2i

- 2mV}2 W .
sin

2o/

if/) be infinitely great compared with q. This condition may be assumed for
the present purpose, because the value of g> (z) is independent of the nature
of the infinite limits in the summation and is therefore unaffected by such a
limitation.

f - "" 1 ] f £+!?(9+1) -*$-* *

l_* J ' l_

The fraction —, has a real part. In the exponent it is multiplied by q + 1.

that is, by an infinite quantity ; so that the real part of the index of
the exponential is infinite, either positive or negative. Thus either the
first term is infinite and the second zero, or vice versa; in either case,

r T i •

sin \z + 2 (q + 1) twl ^— , is infinite, and therefore
2o) J

{2 + 2(q + l)(o- 2m V}2
Similarly for the other sum. Hence

= 0.

In the same way it may be shewn that

£>0 + 2a/)-£>0) = 0;

therefore £> (z) is doubly-periodic in 2<o and 2a>'.

126.] ELLIPTIC FUNCTION 253

Now in any parallelogram whose adjacent sides are 2&> and 2&>', there is
only one infinity and it is of multiplicity two; hence, by § 116, Prop. III.,
Cor. 3, 2o) and 2&>' determine a primitive parallelogram for $> (z).

We shall assume the parallelogram of reference chosen so as to include
the origin.

126. The function $ (z) is thus of the second order and the first class.

Since its irreducible infinity is of the second degree, the only irreducible
infinity of g>' (z} is of the third degree, being the origin ; and the function
<§t ' (z) is odd.

The zeros of jp' (z} are thus &>, ft/, and (&> + to') ; or, if we introduce a new
quantity w" defined by the equation

&>" = &) + &>',

the zeros of <@! (z) are &>, &>', &>".
We take

#>(«) = e1} p(a>") = e2, p(m') = e3, %>(z) = p:
and then, by § 119, Prop. XIV., Cor., we have

where A is some constant. To determine the equation more exactly, we
substitute the expression of jp in the vicinity of the origin. Then

80 that P' = -j+iQff* +

When substitution is made, it is necessary to retain in the expansion all
terms up to z° inclusive. We then have, for |p'2, the expression

4 2 4

and for A (^ - e-,) (p - e2) (p - e3), the expression

A r1 3 9- 3

1L^6 + 20^+285r3+>"

- (e, + e2 + e3)(^+—g2 + ...)+ (6le, + e2e3 + e&) (- + ...)- tf,«A |

When we equate coefficients in these two expressions, we find

e1 + e2 + e3 = 0, e& + e»
therefore the differential equation satisfied by p is

254 PERIODICITY [126.

Evidently £>" = 6§>2 - %gs,

and so on ; and it is easy to verify that the 2wth derivative of g) is a rational
integral algebraical function of <p of degree n + l and that the (2w+l)th
derivative of fp is the product of g>' by a rational integral algebraical function
of degree n.

The differential equation can be otherwise obtained, by dependence on
Cor. 2, Prop. V. of § 116. We have, by differentiation of %>',

for points in the vicinity of the origin ; and also

^+!^2 + r4^2 + "--

Hence <@" and §>2 have the same irreducible infinities in the same degree and
their fractional parts are essentially the same : they are homoperiodic and
therefore they are equivalent to one another. It is easy to see that g>" — 6(jp2
is equal to a function which, being finite in the vicinity of the origin, is finite
in the parallelogram of reference and therefore, as it is doubly -periodic, is
finite over the whole plane. It therefore has a constant value, which can be
obtained by taking the value at any point; the value of the function for

z = 0 is — \g» and therefore

g>"_6^ = -^2,

so that |p"= 6g»2-|<72,

the integration of which, with determination of the constant of integration,

leads to the former equation.

This form, involving the second derivative, is a convenient one by which
to determine a few more terms of the expansion in the vicinity of the origin :
and it is easy to shew that

from which some theorems relating to the sums ^SH"2*1 can be deduced*.

Ex. If cn be the coefficient of 22n~2 in the expansion of $ (z) in the vicinity of the
origin, then

c»=/o~ . iw.. ON 2 Crfin-r- (Weierstrass.)

We have jp'2 = 4^>3 - g$ - g3 ;

the function jjp' is odd and in the vicinity of the origin we have

* See a paper by the author, Quart. Journ., vol. xxii, (1887), pp. 1 — 43, where other references
are given and other applications of the general theorems are made.

126.] OF WEIERSTRASS'S FUNCTION 255

hence, representing by — (4|p3 — g$>— g^ that branch of the function which is
negative for large real values, we have

and therefore z =

The upper limit is determined by the fact that when z = 0, g> = oo ; so that

- r d®

_ r d%>

lp {4 (p - ej (p - e2) (p - e,)}*'

This is, as it should be, an integral with a doubly-infinite series of values.
We have, by Ex. 6 of § 104,

r

0)j = ft) =|

J<h

,

ft>3= ft) =

J

with the relation a)" = a) + co'.

127. We have seen that g> (z) is doubly-periodic, so that

p(*+2») = $>(*),

and therefore dg(5 + 2«) = dgW

a^ dz

hence integrating ?(^ + 2<») = %(z) + A.

Now ^ is an odd function ; hence, taking z — — co which is not an infinity of £,
we have

^ = 2^(&))=27;

say, where r) denotes £ (&>) ; and therefore

£(*-*• 2»)r- £•(*)« 89,

which is a constant.

Similarly %(z + 2&>') - ^ (^) - 2i/,

where r;' = ^ (to') and is constant.

Hence combining the results, we have

% (z + 2w&) + 2?rc V) -£(z) = 2mri + Zmrj',
where m and m are any integers.

It is evident that 77 and rj' cannot be absorbed into £; so that £ is not a
periodic function, a result confirmatory of the statement in § 124.

256 PSEUDO-PERIODIC [127.

There is, however, a pseudo-periodicity of the function £ : its characteristic
is the reproduction of the function with an added constant for an added
period. This form is only one of several simple forms of pseudo-periodicity
which will be considered in the next chapter.

128. But, though %(z) is not periodic, functions which are periodic can
be constructed by its means.

Thus, if 4>(z)=AS(z-a)+Bt(z-V)
then * + 2w-(*) = 2A£(*-a

so that, subject to the condition

A+B+C+...=0,
<j) (z) is a doubly-periodic function.

Again, we know that, within the fundamental parallelograTH, f has a
single irreducible infinity and that the infinity is simple; hence the irre
ducible infinities of the function </> (V) are z = a, b, c, ..., and each is a simple
infinity. The condition A + B + C + ...=0 is merely the condition of Prop.
III., § 116, that the ' integral residue ' of the function is zero.

Conversely, a doubly-periodic function with m assigned infinities can be
expressed in terms of f and its derivatives. Let ax be an irreducible infinity
of <£> of degree n, and suppose that the fractional part of <I> for expansion in
the immediate vicinity of ax is

A! i?i | ^ KI

Then

-if (*- 4).-...

is not infinite for z = a^.

Proceeding similarly for each of the irreducible infinities, we have a
function

r

which is not infinite for any of the points z = alt a2, .... But because <f> (z)
is doubly-periodic, we have

and therefore the function

128.]

FUNCTIONS

257

is doubly-periodic. Moreover, all the derivatives of any order of each of the
functions £ are doubly-periodic; hence the foregoing function is doubly-
periodic.

The function has been shewn to be not infinite at the points a1} a2, ...,
and therefore it has no infinities in the fundamental parallelogram ; con
sequently, being doubly-periodic, it has no infinities in the plane and it is
a constant, say G. Hence we have

g,

r=i

r=i

m

with the condition 2 Ar = 0, which is satisfied because <E> (z) is doubly-
periodic.

This is the required expression * for <I> (z) in terms of the function % and
its derivatives; it is evidently of especial importance when the indefinite
integral of a doubly-periodic function is required.

129. Constants 77 and 77', connected with &> and «', have been introduced
by the pseudo-periodicity of £(z)\ the relation, contained in the following
proposition, is necessary and useful : —

The constants 77, w', &>, &>' are connected by the relation

the + or - sign being taken according as the real part of o>'fa)i is positive or
negative.

A fundamental parallelogram having an angular point at z0 is either of

the form (i) in fig. 34, in which case 9t f-^] is

\mj

positive : or of the form (ii), in which case 9J ( — . )

\Ct)l/

is negative. Evidently a description of the paral
lelogram A BCD in (i) will give for an integral the
'same result (but with an opposite sign) as a de
scription of the parallelogram in (ii) for the same
integral in the direction A BCD in that figure.

We choose the fundamental parallelogram, so
that it may contain the origin in the included
area. The origin is the only infinity of £ which
can be within the area : along the boundary £ is
always finite.

Now since

* See Hermitet Ann. de Toulouse, t. ii, (1888), C, pp. 1—12.
F.

20+2o)'

Fig. 34

258 PSEUDO-PERIODICITY OF WEIERSTRASS's [129.

the integral of £(z) round ABCD in (i), fig. 34, is (§ 116, Prop. II., Cor.)

rD CB

2r)dz - fy'dz,

J A J A

the integrals being along the lines AD and AB respectively, that is, the

integral is

4 (rju>' — rfw}.

But as the origin is the only infinity within the parallelogram, the path of
integration ABCD A can be deformed so as to be merely a small curve round !
the origin. In the vicinity of the origin, we have

and therefore, as the integrals of all terms except the first vanish when taken
round this curve, we have

= 2-Trt.

Hence 4 (rjw — TJ'O)) = 27ri,

and therefore i](f> — rju> = \iri.

This is the result as derived from (i), fig. 34, that is, when 91 [?-) is positive.

\i/tU/

When (ii), fig. 34, is taken account of, the result is the same except
that, when the circuit passes from z0 to z0 + 2&>, then to z0 + 2t» + 2o>',
then to z0 + 2&>' and then to z0, it passes in the negative direction round the'
parallelogram. The value of the integral along the path ABCDA is the
same as before, viz., 4 (rjw — rj'a)) ; when the path is deformed into a small!

rdz

curve round the origin, the value of the integral is I — taken negatively, an

J **

therefore it is — 2?ri : hence

t](£) — rj (a = — \Tri.

Combining the results, we have

rjay' — f]w = ± ^Tri,
/ '\
according: as 9t ( — . ] is positive or negative.

\0)lJ

COROLLARY. If there be a change to any other fundamental parallelo
gram, determined by 2H and 2O', where

£1 = pa) + qa)', £1' = p'co + q'a)',
p, q, p', q' being integers such that pq — p'q = ± 1, and if H, H' denote

C(ft'), then

H = pr} + qrj', H' = p'rj + q'f}' ;

therefore HW - H'£l = ± \-iri,

according as the real part of T^ is positive or negative.

130.] PRODUCT-FUNCTIONS 259

130. It has been seen that £ (z) is pseudo-periodic ; there is also a pseudo-
periodicity for o- (z), but of a different kind. We have

that is,

0-0 +

and therefore a- (z + 2<w) = Ae^zcr (z),

where A is a constant. To determine A, we make z = — &>, which is not a
zero or an infinity of a (z) ; then, since a (z) is an odd function, we have

so that o- 0 + 2&>) = - e*> <z+<0> a- (z).

Hence o-(z + 4eo) = — e*> (z+3ft)) <r(z + 2&>)

and similarly a (z + 2mey) = (— l)m

Proceeding in the same way from

we find a~(z + 2m V) = (- l)m' e*>' <w/z+m'2w') o- (z).

Then a (z + 2ma> + 2m V) = (- l)m e2^ (ww+^»+»»»»V) Q- (^ + 2m/eo/)

== / _ J \m+m' g sz (mij+m'V) +2'?m2<o+47]mmV+27)'m'2co' _.

But lyct)' — r/o) = ± \iri,

SO that g2mm'(r|a>'— rj'w) _ e±mm'iri _ /_ |\nj.m'

and therefore

2m V) = (- l)w

which is the law of change of a (z) for increase of z by integral multiples of
the periods.

Evidently <r(z) is not a periodic function, a result confirmatory of the
statement in § 122. But there is a pseudo-periodicity the characteristic of
which is the reproduction, for an added period, of the function with an
exponential factor the index being linear in the variable. This is another
of the forms of pseudo-periodicity which will be considered in the next
chapter.

131. But though <r(z) is not periodic, we can by its means construct
functions which are periodic in the pseudo-periods of a (z).

By the result in the last section, we have

<r (z — a. + 2ma> + 2m'&)') cr (z — a) + ,,
<r(z-fi + 2mo> + 2m V) ~ <r(z ^~J3) & '

17—2

260 DOUBLY-PERIODIC FUNCTIONS [131.

and therefore, if <f> (z) denote

a- (z — cti) a (z — 02) cr (z — &»)

then $ (z + 2ra&> + 2m V) = e2(m>} +m'*'> <2^-2^

so that $ (z) is doubly-periodic in 2« and 2&>' provided

Now the zeros of <f>(z), regarded as a product of o--functions, are als a2,..., «„
and the points homologous with them ; and the infinities are Pi, /32, ... , ftn and i
the points homologous with them. It may happen that the points a and ft j
are not all in the parallelogram of reference ; if the irreducible points
homologous with them be a1} ..., an and blt ... , bn, then

Sar = ~S.br (mod. 2&>, 2co'),

and the new points are the irreducible zeros and the irreducible infinities of
<}>(z). This result, we know from Prop. III., § 116, must be satisfied.

It is naturally assumed that no one of the points a is the same as, or is
homologous with, any one of the points ft : the order of the doubly-periodic
function would otherwise be diminished by 1.

If any a be repeated, then that point is a repeated zero of <j>(z); similarly-
if any ft be repeated, then that point is a repeated infinity of <£ (z). In every,
case, the sum of the irreducible zeros must be congruent with the sum of the
irreducible infinities in order that the above expression for <j)(z) may be
doubly-periodic.

Conversely, if a doubly-periodic function <£ (z) be required with m assignedJ
irreducible zeros a and m assigned irreducible infinities b, which are subject t

to the congruence

2a = 26 (mod. 2co, 2&>'),

we first find points OL and ft homologous with a and with b respectively sucht
that

rru +U t «

Then the function

a- (z - Pi) a(z — ftm)

has the same zeros and the same infinities as </> (z), and is homoperiodic withi
it ; and therefore, by § 116, IV.,

o-(s-ai) o-^-otm)

9 \z> — •"• „(„ _ o \ *(*—ft V

where A is a quantity independent of z.

Ex. 1. Consider ft? (z). It has the origin for an infinity of the third degree and all thti
remaining infinities are reducible to the origin ; and its three irreducible zeros are a, a/, a>" j
Moreover, since <o"=a>' + a>, we have w + w' + w" congruent with but not equal to zerw
We therefore choose other points so that the sum of the zeros may be actually the same,

131.] EXAMPLES 261

as the sum of the infinities, which is zero ; the simplest choice is to take <», &>', - «".
Hence

where A is a constant. To determine A, consider the expansions in the immediate
vicinity of the origin ; then

2 o- ( - co) <r ( - to ) v (a)")

?"•" ...... S3 * ...... >

sothat y^-g^^/rf^'tf.

O- («) or (eo ) o- (a ) O"3 (2)

Another method of arranging zeros, so that their sum is equal to that of the infinities,
is to take — w, — «', co" ; and then we should find

dy M =2
r W

This result can, however, be deduced from the preceding form merely by changing the
sign of z.

Ex. 2. Consider the function

. a- (u + v) a- (u — v)

*«(«)

where v is any quantity and A is independent of u. It is, qua function of u, doubly-
periodic ; and it has u = 0 as an infinity of the second degree, all the infinities being
homologous with the origin. Hence the function is homoperiodic with g> (u) and it has
the same infinities as $> (u) : thus the two are equivalent, so that

where B and C are independent of u. The left-hand side vanishes if n—v; hence
(v), and therefore

where A' is a new quantity independent of u. To determine .4' we consider the
expansions in the vicinity of u = 0 ; we have

A.'<r(v)<r(-v)

sothat

and therefore cr- =

o-2 (%) o-2 (v)

a formula of very great importance.

Ex. 3. Taking logarithmic derivatives with regard to u of the two sides of the last
equation, we have

and, similarly, taking them with regard to D, we have

whence

262 EXAMPLES [131.

giving the special value of the left-hand side as (§ 128) a doubly-periodic function. It is
also the addition-theorem, so far as there is an addition-theorem, for the ^-function.

Ex. 4. We can, by differentiation, at once deduce the addition-theorem for g) (u + v).
Evidently

which is only one of many forms : one of the most useful is

which can be deduced from the preceding form.

The result can be used to modify the expression for a general doubly-periodic function
* (z) obtained in § 128. We have

Each derivative of f can be expressed either as an integral algebraical function of $ (z - a,.)
or as the product of jjp' (z — ar) by such a function ; and by the use of the addition-theorem
these can be expressed in the form

L >
where L, M, N are rational integral algebraical functions of $(z). Hence the function

can be expressed in the same form, the simplest case being when all its infinities are
simple, and then

4. (z) = C+ 2 Ar{(e-ar)

(*)-§» (Or)

with the condition 2 Ar = 0.

r=l

Ex. 5. The function $ (z) — e1 is an even function, doubly-periodic in 2« and 2o> and
having 2 = 0 for an infinity of the second degree ; it has only a single infinity of the second
degi'ee in a fundamental parallelogram.

Again, z = &> is a zero of the function ; and, since ^X («) = 0 but $>" (o>) is not zero, it is a
double zero of $ (z)-el. All the zeros are therefore reducible to 2 = o> ; and the function
has only a single zero of the second degree in a fundamental parallelogram.

Taking then the parallelogram of reference so as to include the points 0 = 0 and 0=o>,
we have

where Q (z) has no zero and no infinity for points within the parallelogram.

Again, for g> (z + o>) - e± , the irreducible zero of the second degree within the parallelo-

131.] OF DOUBLY-PERIODIC FUNCTIONS 263

gram is given by S + <B = O>, that is, it is 0 = 0; and the irreducible infinity of the second
degree within the parallelogram is given by z + a = 0, that is, it is z = v. Hence we have

where Ql (z) has no zero and no infinity for points within the parallelogram.

Hence {£> (z} - ej {%> (z + «) - ej m Q (z) Q1 (z\

that is, it is a function which has no zero and no infinity for points within the
parallelogram of reference. Being doubly-periodic, it therefore has no zero and no infinity
anywhere in the plane ; it consequently is a constant, which is the value for any point.
Taking the special value s = a>, we have jp(m') = es, and (jf>(a>' + a>) = e2 ; and therefore

{#> (*) ~ e,} (V (* + «)- e,} = (e3 - *i) (e, - *i).

Similarly {#> (z) - e2} {#> (z + »") - e2} = (ex - e2) (es - e2),

and {#> (2) - ^ {p (z + a)') - e3} = (e2 - e3} (^ - <?3).

It is possible to derive at once from these equations the values of the ^-function for
the quarter-periods.

Note. In the preceding chapter some theorems were given which indicated that
functions, which are doubly-periodic in the same periods, can be expressed in terms of one
another : in particular cases, care has occasionally to be exercised to be certain that the
periods of the functions are the same, especially when transformations of the variables are
effected. For instance, since g) (z) has the origin for an infinity and sn u has it for a zero,
it is natural to express the one in terms of the other. Now $ (z) is an even function, and
sn u is an odd function ; hence the relation to be obtained will be expected to be one
between ®(z) and sn2w. But one of the periods of sn2 u is only one-half of the correspond
ing period of sn u ; and so the period-parallelogram is changed. The actual relation* is

(P (z) - <?3 = (<?!- e3) sn-2«,
where u = (el -esf z and F = (<?2 -e^)l(el -e:i).

Again, with the ordinary notation of Jacobian elliptic functions, the periods of sn z are
4 A" and 2iA", those of dn z are 2 A and 4i K', and those of en z are 4 A' and 2A'+ 2iA''. The
squares of these three functions are homoperiodic in 2K and ZiK' ; they are each of the
second order, and they have the same infinities. Hence sn2 z, en2 z, dn2 z are equivalent to
one another (§ 116, V.).

But such cases belong to the detailed development of the theory of particular classes of
functions, rather than to what are merely illustrations of the general propositions.

132. As a last illustration giving properties of the functions just
considered, the derivatives of an elliptic function with regard to the periods
will be obtained.

Let (/> (z) be any function, doubly-periodic in 2o> and 2&/ so that

</> (z + 2m&> + 2m V) = </> (z),

the coefficients in <f> implicitly involve &> and CD'. Let <f>1} <£2, and </>' respec
tively denote 90/3t», 9<£/9o/, 9^/9^ ; then

^ (z + 2771&) + 2m' to') + %m<j>' (z + 2mo> + 2wV) = fa (z),
fa (z + 2m&) + 2??iV) -1- 2m'<£' (z + 2rao> + 2wV) = fa (z),

$ (z + 2m&) + 2m V) = <f> (z).

" Halphen, Fonctions Elliptiques, t. i, pp. 23 — 25.

264 PERIOD-DERIVATIVES [132.

Multiplying by &>, ro', z respectively and adding, we have
&></>! (z + 2mo> + 2m V) + o>'</>2 (z + 2m<« + 2mV)

+ (z + 2mw + 2wV) </>' 0 + 2mo)

= 0)0! (Z) + 0)'(f).2 (Z) + Z<f>' (Z).

Hence, if f(z} = mfa (z) + 6/02 (z) + z$ (z),

then f(z) is a function doubly -periodic in the periods of (f>.

Again, multiplying by rj, 77', %(z), adding, and remembering that
£0 + 2mm + 2raV) = £($)

we have

77$! (z + 2mw + 2m V) + 7/<£2 0 + 2m«o + 2m V)

+ £(z + 2mm + 2m' ay') <f>' (z + 2mw + 2m'a>')

-ik<fi)+J+,(*) + f(*Wto

Hence, if g (z) = yfa (z) + q'fa (z) + £ (z) $' (z),

then g(z) is a function doubly-periodic in the periods of <f>.

In what precedes, the function <f>(z) is any function, doubly-periodic in
2o>, 2&)' ; one simple and useful case occurs when 0 (z) is taken to be the
function z. Now

and fW-J-^

hence, in the vicinity of the origin, we have

9P >d@ d& 2

(o ^- + 03 5*7 + jp-«- as — + even integral powers of z1
d(o d(o dz z-

= -2^>,

since both functions are doubly-periodic and the terms independent of z
vanish for both functions. It is easy to see that this equation merely
expresses the fact that <p, which is equal to

l

is homogeneous of degree — 2 in z, &>, to'.
Similarly

9|J> / ty ^d%> 22

77 2+*) ~r-/ + b (-2r) y- = - ~i + j^ 9-i + even integral powers of z.

But, in the vicinity of the origin,

,5-7 = — + YQ^ -I- even integral powers of 2,

132.] OF WEIEHSTRASS'S FUNCTION 265

so that

9P / d@ »/ \ dP 1 32lP • x r

17 3^- + V g :S + f<«) |^t i gji ™i** even mtegral powers of z.

The function on the left-hand side is doubly-periodic : it has no infinity
at the origin and therefore none in the fundamental parallelogram ; it there
fore has no infinities in the plane. It is thus constant and equal to its value
anywhere, say at the origin. This value is ^gz> and therefore

T/w's equation, when combined with

,

+ eo' ; + z = - ty,
dco da> oz

j

gives the value of ~- and -^ , .
J dm 9&)

The equations are identically satisfied. Equating the coefficients of z2 in
the expansions, which are valid in the vicinity of the origin, we have

and equating the coefficients of ^ in the same expansions, we have

Hence for any function u, which involves w and &/ and therefore implicitly
involves g2 and ^r3, we have

du ,du
w 5- + w — , =

aw a&)

9w . , 3w
17 a- +T;' — = -
9&) 9&)

Since ^) is such a function, we have

f : *'

being ^/te equations which determine the derivatives of $ with regard to the
invariants g., and g.^.

266 EVEN [132.

The latter equation, integrated twice, leads to
9V da- 2 80- 1

a differential equation satisfied by <r(z)*.

133. The foregoing investigations give some of the properties of doubly-
periodic functions of the second order, whether they be uneven and have two
simple irreducible infinities, or even and have one double irreducible infinity.

If a function U of the second order have a repeated infinity at z = y, then
it is determined by an equation of the form

or, taking U - £ (X + fi + v) = Q, the equation is

Q'» = 4a2 [(Q - e,) (Q - e,) (Q - $,)]*,

where ^ + e2 + e3 = 0. Taking account of the infinities, we have

Q=@(az- ay) ;

and therefore U-±(\ + /Ji. + v) = %> (az - ay)

. . 1 (tp'(az) + cp (ay)}'2
= -Q (az) - <o (ay) + — —

a x ' o x "

by Ex. 4, p. 262. The right-hand side cannot be an odd function; hence
an odd function of the second order cannot have a repeated infinity. Similarly,
by taking reciprocals of the functions, it follows that an odd function of the
second order cannot have a repeated zero.

It thus appears that the investigations in §§ 120, 121 are sufficient for the
included range of properties of odd functions. We now proceed to obtain
the general equations of even functions. Every such function can (by § 118,
XIII., Cor. I.) be expressed in the form |a#> (z) +b}+ {c#> (z) + d], and its
equations could thence be deduced from those of p(z)\ but, partly for
uniformity, we shall adopt the same method as in § 120 for odd functions.
And, as already stated (p. 251), the separate class of functions of the second
order that are neither even nor odd, will not be discussed.

134. Let, then, <j>(z) denote an even doubly-periodic function of the
second order (it may be either of the first class or of the second class) and let
2&), 2<w' be its periods ; and denote 2&) + 2ft)' by 2o>". Then

since the function is even ; and since

<£ (ft) + Z) = <f> (— &) — z)

= <f> (2&) — &) — z)
= (j) (CD — *),

* For this and other deductions from these equations, see Frobenius und Stickelberger, Crelle,
t. xcii, (1882), pp. 311—327; Halphen, Traite des feme t ions elliptiqucs, t. i, (1886), chap. ix. ;
and a memoir by the author, quoted on p. 254, note.

134.] DOUBLY-PERIODIC FUNCTIONS

it follows that <£ (&> + z)— and, similarly, $ (&>' + z) and 0 (to" + z) are even
functions.

Now </> (w + a), an even function, has two irreducible infinities, and is
periodic in 2&>, 2&/ ; also <£ (z), an even function, has two irreducible infinities
and is periodic in 2&>, 2&/. There is therefore a relation between 0 (z) and
</> (w +z), which, by § 118, Prop. XIII., Cor. I., is of the first degree in <£ (z) and
of the first degree in <j) (&> + z) ; thus it must be included in

B<f> (z) <j>(<o + z)-C<l> (z) -C'<t>(a> + z)+A = 0.

But <£ (z) is periodic in 2<w ; hence, on writing z + <w for z in the equation, it

becomes

B<f>(a>+z)<j>(z)-C<f>(co+z)-C'<l>(z) + A=0;

thus tf=C".

If B be zero, then (7 may not be zero, for the relation cannot become
evanescent : it is of the form

A' .............................. (1).

If B be not zero, then the relation is

Treating <f> (w + z) in the same way, we find that the relation between it

and (f) (z) is

F(j> (z) (f> (ay' + z)-D(j> (z} -D(j>(a>' + z) + E = 0,

so that, if F be zero, the relation is of the form

£(*) + 0(a>' + *) = J0' ........................... (I)',

and, if F be not zero, the relation is of the form

Four cases thus arise, viz., the coexistence of (1) with (1)', of (1) with (2)',
of (2) with (1)', and of (2) with (2)'. These will be taken in order.

I. : the coexistence of (1) with (1)'. From (1) we have

<j> (a> + z} + (j> (&>" + z) = A',

so that </> (z) + <f) (w + z) + (f) (w + z) + 0 (<w" + z) = 2A'.

Similarly, from (1)',

so that A = E', arid then

(f)((0 + z)=(j)((o'+ Z\

whence <w ~ &>' is a period, contrary to the initial hypothesis that 2&> and 2&>'
determine a fundamental parallelogram. Hence equations (1) and (1)' cannot
coexist.

268 EVEN [134.

II. : the coexistence of (1) with (2)'. From (1) we have
<£(«" + z} = A' - <£(&>' + z)

on substitution from (2)'. From (2)' we have

cb (co + z) = -5*1) -- ( — =r
F<f) (CD + z) - D

_ (A'D -E)-D<j> (z)
= A'F - D - F(f> (z) '
on substitution from (1). The two values of <£ (&>" + z) must be the same,

whence

A'F-D = D,

which relation establishes the periodicity of </> (z) in 2ft)", when it is considered
as given by either of the two expressions which have been obtained. We

thus have

A'F=W-
and then, by (1), we have

<f>(z)-j+<l>(
and, by (2)', we have

If a new even function be introduced, doubly-periodic in the same periods
having the same infinities and defined by the equation

& 0) = </> 0) - J >
the equations satisfied by fa (z) are

fa(a> + z) + fa(z) = 0 }

fa (&)' + z) fa (z) = constant] '

To the detailed properties of such functions we shall return later ; meanwhile
it may be noticed that these equations are, in form, the same as those satisfied
by an odd function of the second order.

III. : the coexistence of (2) with (1)'. This case is similar to II., with the
result that, if an even function be introduced, doubly-periodic in the same
periods having the same infinities and defined by the equation

C

fa (Z) = <£ (2) - -g ,

the equations satisfied by fa (z) are

fa (&>' + z) + fa (z) = 0 }

fa (&) + z) fa (z) = constant] '

It is, in fact, merely the previous case with the periods interchanged.

134.] DOUBLY-PERIODIC FUNCTIONS 269

IV. : the coexistence of (2) with (2)'. From (2) we have

_ (CD - AF) <ft (z) - (GE - AD)
~ (BD - CF) <J> (z) - (BE - CD) '

on substitution from (2)'. Similarly from (2)', after substitution from (2), we
have

~

The two values must be the same ; hence

CD-AF=-(GD-BE\

which indeed is the condition that each of the expressions for <ft (&>" + z)
should give a function periodic in 2&>". Thus

One case may be at once considered and removed, viz. if C and D vanish
together. Then since, by the hypothesis of the existence of (2) and of (2)',
neither B nor F vanishes, we have

A__E

B~ F'

so that u + , =

and then the relations are <£ (&> + z) + <f) (&>' + z) = 0,

or, what is the same thing, <ft (Y) + <ft (&>" + z) — 0 ]

and </> (z) </>(&> + z) = constant j '

This case is substantially the same as that of II. and III., arising merely

from a modification (§ 109) of the fundamental parallelogram, into one whose

sides are determined by 2&> and 2&>".

Hence we may have (2) coexistent with (2)' provided

AF + BE=WD;
C and D do not both vanish, and neither B nor F vanishes.

IV. (1). Let neither C nor D vanish ; and for brevity write

<f>((o + z)=<l>1, (f> (w" + z) = <£o, </> (&)' + z) = $3, (f) (z) = 0.
Then the equations in IV. are

Now a doubly-periodic function, with given zeros and given infinities, is
determinate save as to an arbitrary constant factor. We therefore introduce
an arbitrary factor X, so that

<£=Xi/r,

G D

and then taking = CI' = Ca'

270 EVEN [134.

£
we have (^ - Cl) (fa - Cj) = d2- -^ ,

ET

The arbitrary quantity A, is at our disposal : we introduce a new quantity c2,
defined by the equation

A

Tt-. o — Ci (C2 + €3) C2C3 ,

and therefore at our disposal. But since

AF + BE=2CD,

A E .CD

we have ^ + ^ = 2 ^ ^ = 2Clc3j

ri

and therefore ^--2 = c3 (Cj + c2) - 0^2 .

Hence the foregoing equations are

- d) = (Cj - C2) (d - C3),

- C3) = (C3 - d) (C3 - C2).

The equation for ^>2, that is <f>((o" + z), is

_Lcf)-M

where L= CD - BE = AF - CD, M=AD-CE, N=CF-BD,

so that ^ + 5M" = 2CL.

As before, one particular case may be considered and removed. If N be

zero, so that

C_D_

B~F~a

AE CD ,.

say, and B+F=RF= '

then we find $ + ^>2 = ^>i + <£3 = 2«,

or taking a function ^ = 0 — a,

the equation becomes % (^ + % C^" + ^ = 0.
The other equations then become

and therefore they are similar to those in Cases II. and III.
If N be not zero, then it is easy to shew that
N=BF\(c1-c3)>

M = BF\3 (d - c3) (c.,C! + c,c3 - dc3) ;

134.] DOUBLY-PERIODIC FUNCTIONS 271

and then the equation connecting 0 and 02 changes to

s - Ca) = (Ca - Cx) (Ca - Cs)

which, with (^ — d) ("^i — d) = (d — c2) (d — c3)

( r — ^3/ \ i 3 — ^3/ == V^3 ^" ^3 ^2'

are relations between ty, ^rl} -^2, ty.3, where the quantity c2 is at our disposal.

IV. (2). These equations have been obtained on the supposition that
neither G nor D is zero. If either vanish, let it be C: then D docs not
vanish ; and the equations can be expressed in the form

E

D\

J

E\ E(D*-EF)

We therefore obtain the following theorem :

If (f> be an even function doubly-periodic in 2&> and 2&>' and of the second
order, and if all functions equivalent to <J> in the form R<f> + 8 (where R and
S are constants) be regarded as the same as 0, then either the function satisfies
the system of equations

00) 0O"
where H is a constant ; or it satisfies the system of equations

{0 0) - d} {0 (ft) +Z)- d] = (Ci - C2) (d - C3)
{00)-C3}{0(>/ +^)-C3}=(Cs-C1)(Cs-Ca)
{0 0) - C2} (0 ((,)" + Z)- C2} = (C2 ~ d) (C2 - Cs)

where of the three constants clt c2, cs one can be arbitrarily assigned.
We shall now very briefly consider these in turn.

135. So far as concerns the former class of equations satisfied by an even
doubly-periodic function, viz.,

we proceed initially as in (§ 120) the case of an odd function. We have the
further equations

00) = 0(-4

0 (ft) + Z) — 0 (ft) — Z), 0 (a/ + Z) = 0 (ft)' — Z).

* The systems obtained by the interchange of w, w', w" among one another in the equations
are not substantially distinct from the form adopted for the system I. ; the apparent difference
can be removed by an appropriate corresponding interchange of the periods.

272 EVEN DOUBLY-PERIODIC FUNCTIONS [135.

Taking z = — ^w, the first gives

so that ^&> is either a zero or an infinity.
If \<£> be a zero, then

(f> (f to) = $ (<« + ^ft)) = — <f> (^») by the first equation

= 0,
so that ^&> and f&> are zeros. And then, by the second equation,

&)' + ^<w, &)' 4- f a)
are infinities.

If \w be an infinity, then in the same way |w is also an infinity ; and
then a)' + \w, &>' + f &) are zeros. Since these amount merely to interchanging
zeros and infinities, which is the same functionally as taking the reciprocal of
the function, we may choose either arrangement. We shall take that which
gives ^0), f &> as the zeros ; and &>' 4- ^&>, &/ + f &> as the infinities.

The function <j> is evidently of the second class, in that it has two distinct
simple irreducible infinities.

Because &>' + |&), &>' + f &> are the irreducible infinities of </> (z), the four
zeros of $' (z} are, by § 117, the irreducible points homologous with &>",
&)" + &>, &>" + a)', a)" + &)", that is, the irreducible zeros of (f)' (z) are 0, &>, &>', &>".
Moreover

by the first of the equations of the system ; hence the relation between (f> (
and ((>' (z) is

#* (z} = A{<t>(z)-$ (())} {</> (z) - (/> («)} |0 (*) - (/> (ft)')} {(/> (*) - </> («")}

= A [p (0) - p (z)}{p (ft)') - ^ (*)}.
Since the origin is neither a zero nor an infinity of <£ (^), let

so that </>j (0) is unity and 0/ (0) is zero ; then

^(*)«X»{l-^(*)}{^-^(f))

the differential equation determining fa (z).

The character of the function depends upon the value of p and the
constant of integration. The function may be compared with en u, by taking

2ft), 2&/ = 4>K, 2K + 2iK' ; and with — *— , by taking 2ft), 2ft)' = 2K, MK',

dn u

which (§ 131, note) are the periods of these (even) Jacobian elliptic functions.
We may deal even more briefly with the even, function characterised by
the second class of equations in § 134. One of the quantities c1} c2, c3 being
at our disposal, we choose it so that

Ci + c2 + c3 = 0 ;

and then the analogy with the equations of Weierstrass's ^-function is
complete (see § 133).

CHAPTER XII.

PSEUDO-PERIODIC FUNCTIONS.

136. MOST of the functions in the last two Chapters are of the type
called doubly-periodic, that is, they are reproduced when their arguments are
increased by integral multiples of two distinct periods. But, in §§ 127, 130,
functions of only a pseudo-periodic type have arisen : thus the ^-function
satisfies the equation

m2&> + m'2&>') = £(» + m2i) + m'2v',
,nd the cr-function the equation

m' i (mr,+m'r,') (z+wuo+m'oi1)

These are instances of the most important classes: and the distinction
between the two can be made even less by considering the function
e^(z} — ^(z), when we have

£ (z + ra2&> + m'2&>') = e-mr> e"™'*' % (z).

In the case of the ^-function an increase of the argument by a period leads
to the reproduction of the function multiplied by an exponential factor that
is constant, and in the case of the <r-function a similar change of the
argument leads to the reproduction of the function multiplied by an
exponential factor having its index of the form az + b.

Hence, when an argument is subject to periodic increase, there are three
simple classes of functions of that argument.

First, if a function f(z) satisfy the equations

/(* + 2fi>) =/(*), /(* + 2«') =/(*),

it is strictly periodic : it is sometimes called a doubly-periodic function of the
first kind. The general properties of such functions have already been
considered.

Secondly, if a function F(z) satisfy the equations

F (z + 2&>) = pF (z), F (z + 2&/) - pfF (z),
F- 18

274 THREE KINDS [136.

where /u, and fjf are constants, it is pseudo-periodic : it is called a doubly-
periodic function of the second kind. The first derivative of the logarithm
of such a function is a doubly-periodic function of the first kind.

Thirdly, if a function <f> (z) satisfy the equations

<j>(z + 2o)) = eaz+b <j> (z\ <f>(z + 2ft)') = ea'z+v (j> (z),

where a, b, a', b' are constants, it is pseudo-periodic : it is called a doubly-
periodic function of the third kind. The second derivative of the logarithm
of such a function is a doubly-periodic function of the first kind.

The equations of definition for functions of the third kind can be
modified. We have

. <f> (Z + 2ft) + 2ft)') = e«(2+2<o')+6+a'z+6' <£ (z)
— ga' (2+2o>) +b'+az+b J, fz\

whence a'oo — am' = — nnri,

where ra is an integer. Let a new function E (z) be introduced, defined by

the equation

£(«)«*"+*• t(*)i

then X and /A can be chosen so that E (z} satisfies the equations

E(z + 2a)) = E (z\ E(z+ 2ft)') = eAz+£ E (z\
From the last equations, we have

E (z + 2&) + 2ft)') = eA(*+**+B E ^

= eAz+s E (z),
so that 2Aa) is an integral multiple of 2?™'.

Also we have E(z + 2o>) = e*(*-*»'+^+a») <j>(z + 2o>)

so that 4X&) + a = 0,

and 4A,ftr + 2/A&) +6 = 0 (mod.

Similarly, E (z + 2ft)') = e^+wj'+^+w, 0 ^ + 2ft)')

so that 4Xo)' + a = A,

and 4W2 + 2/^co' + 6' = B (mod. 27ri).

From the two equations, which involve X and not //,, we have

Aco = a'o) — aw'

agreeing with the result with 2 A co is an integral multiple of Ziri.

And from the two equations, which involve /j,, we have, on the elimination
of /j, and on substitution for X and A,

b'co — 6ft)' — a&)' (ft)' — &)) = 5ft) (mod. 2-Tn').

136.] OF DOUBLY-PERIODIC FUNCTIONS 275

If A be zero, then E(z) is a doubly-periodic function of the first kind
when eB is unity, and it is a doubly-periodic function of the second kind
when eB is not unity. Hence A, and therefore m, may be assumed to be
different from zero for functions of the third kind. Take a new function
3?z such that

mm
then <l> (z) satisfies the equations

4) (z + 2&)) = <I> (z\ <&(z + 2o)') = e w 3>(z)

* / \ /' \ / \ /t

which will be taken as the canonical equations defining a doubly -periodic
function of the third kind.

Ex. Obtain the values of X, p, A, B for the Weierstrassian function ir(z).

We proceed to obtain some properties of these two classes of functions
which, for brevity, will be called secondary-periodic functions and tertiary-
periodic functions respectively.

Doubly-Periodic Functions of the Second Kind.

For the secondary-periodic functions the chief sources of information are

Hermite, Comptes Rendus, t. liii, (1861), pp. 214—228, ib., t. Iv, (1862), pp. 11—18,
85 — 91 ; Sur quelques applications des fonctions elliptiques, §§ I — in, separate
reprint (1885) from Comptes Rendus; "Note sur la theorie des fonctions ellip
tiques" in Lacroix, vol. ii, (6th edition, 1885), pp. 484—491; Cours d' Analyse,
(4me ed.), pp. 227—234.

Mittag-Leffler, Comptes Rendus, t. xc, (1880), pp. 177 — 180.

.Frobenius, Crelle, t. xciii, (1882), pp. 53 — 68.

Brioschi, Comptes Rendus, t. xcii, (1881), pp. 325—328.

Halphen, Traite' des fonctions elliptiques, t. i, pp. 225 — 238, 411 — 426, 438 442, 463.

137. In the case of the periodic functions of the first kind it was proved
that they can be expressed by means of functions of the second order in the
same period — these being the simplest of such functions. It will now be
proved that a similar result holds for secondary- periodic functions, defined by
the equations

Take a function Q (z} =

a (z) a- (a)

then we have G(z+2a>) = <r(* + g

a (a) a (z + 2w)

arid G(z+ 2&/) = e'V«+2W Q. (^).

The quantities a and X being unrestricted, we choose them so that

„ _ g2rja+2A<o ' __ g2T)'a+2A(o' •

and then G (z), a known function, satisfies the same equation as F (z).

18—2

276 PSEUDO-PERIODIC FUNCTIONS [137.

Let u denote a quantity independent of z, and consider the function

f(Z) = F(z)G(u-z}.
We have f(z + 2o>) = F(z + 2o>) G (u- z - 2w)

=/(*) ;

and similarly f(z + 2<o') =f(z),

so that/(X) is a doubly-periodic function of the first kind with 2« and 2o>'

for its periods.

The sum of the residues of f(z) is therefore zero. To express this sum,
we must obtain the fractional part of the function for expansion in the
vicinity of each of the (accidental) singularities of f(z), that lie within the
parallelogram of periods. The singularities of/ (2) are those of G (u — z) and
those of F(z).

Choosing the parallelogram of reference so that it may contain u, we have
z = u as the only singularity of G (u — z) and it is of the first order, so that,
since

$(£) — =+ positive integral powers of f
in the vicinity of £= 0, we have, in the vicinity of u,
f(z) = {F (u) + positive integral powers of u — z} \ — -4- positive powers I

= -- — + positive integral powers of z — u ;

hence the residue of/(Y) for u is —F(u}.

Let z = c be a pole of F (z) in the parallelogram of order n + 1 ; and, in
the vicinity of c, let

(?! _ cf / 1 \ „ dn ( 1 \

F(z) = ^—c +G^Z (jr^J + • • • + C'n+i fan (zITc) + P°sltlve integral powers.

Then in that vicinity

and therefore the coefficient of - in the expansion of f(z) for points in the

Z ~~ 0

vicinity of c is

which is therefore the residue off(z) for c.

This being the form of the residue of f(z) for each of the poles of F (z),
then, since the sum of the residues is zero, we have

137.] OF THE SECOND KIND 277

or, changing the variable,

,. ..n+l n -

where the summation extends over all the poles of F(z) within that parallelo
gram of periods in which z lies. This result is due to Hermite.

138. It has been assumed that a and \, parameters in 0, are determinate,
an assumption that requires /j, and ^ to be general constants : their values
are given by

yd 4- &>X = | log fjb, r)'a + &/X — \ log //,

and, therefore, since ijca' — rfca = ± ^ITT, we have

+ ITTCL = w' log /JL — co log //)

+ iir\ = — V) log /i + 77 log /z'j '

Now X may vanish without rendering G (z) a null function. If a vanish (or,
what is the same thing, be an integral combination of the periods), then G (z)
is an exponential function multiplied by an infinite constant when X does not
vanish, and it ceases to be a function when X does vanish. These cases must
be taken separately.

First, let a and X vanish* ; then both //, and /// are unity, the function F
is doubly-periodic of the first kind ; but the expression for j^is not determinate,
owing to the form of G. To render it determinate, consider X as zero and a
as infinitesimal, to be made zero ultimately. Then

„,, o-(z) + aa'(z) + ... .^

(*(z) = - - ~ — — (1 + positive integral powers of a)

= - + £ (z) + positive powers of a.
a

Since a is infinitesimal, /JL and /j,' are very nearly unity. When the
function F is given, the coefficients C1} <72, ... may be affected by a, so that
for any one we have

Ck — bk + ayk + higher powers of a,

where yh is finite ; and bk is the actual value for the function which is strictly
of the first kind, so that

Sk-O,

the summation being extended over the poles of the function. Then retaining
only a"1 and a°, we have

This case is discussed by Hermite (I.e., p. 275).

278 MITTAG-LEFFLER'S THEOREM [138.

where C0, equal to £71, is a constant and the term in - vanishes. This expres-

CL

sion, with the condition S^ = 0, is the value of F (u) or, changing the variables,
we have

with the condition S&i = 0, a result agreeing with the one formerly (§ 128)
obtained.

When F is not given, but only its infinities are assigned arbitrarily, then
SO = 0 because F is to be a doubly-periodic function of the first kind ; the

term - "£C vanishes, and we have the same expression for F(z) as before.
Secondly, let a vanish* but not \, so that ^ and // have the forms

We take a function g (z) =

then g(z- 2o>) = ^ e^ £ (z - 2eo )

and g(z-2a>') = p'-1 {g (z} - 2?/ e^} .

Introducing a new function H (z) defined by the equation

we have H (z + 2t») = H (z) - 2ijeA <«-*» F (z),

and H (z + 2o>') = H (z) - 27?V<M-*> F(z).

Consider a parallelogram of periods which contains the point u ; then, if © be

the sum of the residues of H (z) for poles in this parallelogram, we have

the integral being taken positively round the parallelogram. But, by § 116,
Prop. II. Cor., this integral is

f e-*(p+*-«) F (p + 2tot) dt - 0/77 f e-^+
Jo Jo

where p is the corner of the parallelogram and each integral is taken for real
values of t from 0 to 1. Each of the integrals is a constant, so far as concerns
u ; and therefore we may take

® = -Ae^u,

the quantity inside the above bracket being denoted by —\i-rrA.

The residue of H (z) for z = u, arising from the simple pole of g (u — z), is
-F(u) as in§ 137.

If z = c be an accidental singularity of F (z) of order n+1, so that, in the
vicinity of z = c,

F(.) = C, + 0. A- + . . . + BU i- + P (, - c),

This is discussed by Mittag-Leffler, (I.e., p. 275).

138.] ON SECONDARY FUNCTIONS 279

then the residue of H (z) for z = c is

d dn

and similarly for all the other accidental singularities of F (z}. Hence

F(z) = A** +

where the summation extends over all the accidental singularities of F (z) in a
parallelogram of periods which contains z, and y (z) is the function exz%(z}.
This result is due to Mittag-Leffler.

Since /* = e2*" and

g (z - c + 2&>) = fig (z - c) +
we have

and therefore 2 (Gl + C.2\ 4- . . . + Gn+l\n) e~^ = 0,

the summation extending over all the accidental singularities of F(z). The

same equation can be derived through ^F(z) = F(z + 2&>').

Again 2(7: is the sum of the residues in a parallelogram of periods, and
therefore

the integral being taken positively round it. If p be one corner, the integral

n

F (p + 2co't) dt,

Jo

IS

/•i n

o
, each integral being for real variables of t.

Hermite's special form can be derived from Mittag-Leffler's by making \
vanish.

Note. Both Hermite and Mittag-Leffler, in their investigations, have
used the notation of the Jacobian theory of elliptic functions, instead of
dealing with general periodic functions. The forms of their results are as
follows, using as far as possible the notation of the preceding articles.

I. When the function is denned by the equations

F (z + 2K) = ^F (z), F(z+ 2iK') = ^F (z),

then F(z) =

280 INFINITIES AND ZEROS [138.

(the symbol H denoting the Jacobian .ff-function), and the constants <w and X
are determined by the equations

II. If both X and to be zero, so that F(z) is a doubly-periodic function
of the first kind, then

with the condition 5$i = 0.

III. If W be zero, but not X, then

...
where g (z} = --& V,

the constants being subject to the condition

2 (G, + C,\ + . . . + Gn+1 X")e-Ac = 0,

and the summations extending to all the accidental singularities of F(z) in a
parallelogram of periods containing the variable z.

139. Reverting now to the function F(z) we have G (z), defined as

a (z) a (a)

when a and X are properly determined, satisfying the equations
G(z + 2a>) = ftG (z), £0 + 2&/) = yu/£0).

Hence H (z) = F(z)/G (z) is a doubly-periodic function of the first kind ; and
therefore the number of its irreducible zeros is equal to the number of its
irreducible infinities, and their sums (proper account being taken of multipli
city) are congruent to one another with moduli 2« and 2&>'.

Let Ci, c2,..., cm be the set of infinities of F (z) in the parallelogram of
periods containing the point z ; and let y:, . . . , 7^ be the set of zeros of F (z) in
the same parallelogram, an infinity of order n or a zero of order n occurring
n times in the respective sets. The only zero of 0 (z) in the parallelogram is
congruent with — a, and its only infinity is congruent with 0, each being
simple. Hence the m+l irreducible infinities of H (z) are congruent with

a, GI, GZ, . . . , cm,
and its /* + 1 irreducible zeros are congruent with

0, 71, 7s> •••>%*;
and therefore m + 1 = p, + 1,

139.] OF SECONDARY FUNCTIONS 281

From the first it follows* that the number of infinities of a doubly-periodic
function of the second kind in a parallelogram of periods is equal to the number
of its zeros, and that the excess of the sum of the former over the sum of the
latter is congruent with

, (°>' i w i ,
+ — , log it -. log u,

- \7Tl TTl 6 '

/

the sign being the same as that of 9t ( —

\10)

The result just obtained renders it possible to derive another expression
for F (z), substantially due to Hermite. Consider a function

F (Z) = °-Q-7i) 0-0-72).. -0-0-7™) ePZ

(T(z-c1)(r(z-c2)...ar(z-cm) '

where p is a constant. Evidently F1 (z) has the same zeros and the same
infinities, each in the same degree, as F (z). Moreover

F, (Z + 2ft)) = Fl (Z) e2,(2C-2y) + 2pWj
F1 (Z + 2ft)') = F! (Z) e2V(2e-2y)+2P«,'t

If, then, we choose points c and 7, such that

Sc — £7 = a,
and we take p = \ where a and X are the constants of G (z), then

F, (z + 2co) = ^ (z), F, (z + 2ft>') = n'Fj. (z).

The function Fl (z)/F(z) is a doubly-periodic function of the first kind and by
the construction of Fl (z) it has no zeros and no infinities in the finite part of
the plane: it is therefore a constant. Hence

F(z] = A gfr-'ftM*- •/»)•••* (*—*») ^

a(z- c,) a- (z - C2). . .o- (z - Cm)

where Sc — £7 = a, and a and A, are determined as for the function G (z}.

140. One of the most important applications of secondary doubly-periodic
functions is that which leads to the solution of Lame's equation in the cases
when it can be integrated by means of uniform functions. This equation is
subsidiary to the solution of the general equation, characteristic of the
potential of an attracting mass at a point in free space; and it can be
expressed either in the form

jY = (Ak'2 sn2 z + B) w,
or in the form - 2 - = (A@ (z) + B} w,

* Frobenius, Crelle, xciii, pp. 55 — 68, a memoir which contains developments of the properties
of the function G (z). The result appears to have been noticed first by Brioschi, (Comptes Ilendus,
t. xcii, p. 325), in discussing a more limited form.

282 LAMP'S [140.

according to the class of elliptic functions used. In order that the integral
may be uniform, the constant A must be n (n -f 1), where n is a positive
integer ; this value of A, moreover, is the value that occurs most naturally in
the derivation of the equation. The constant B can be taken arbitrarily.

The foregoing equation is one of a class, the properties of which have
been established* by Picard, Floquet, and others. Without entering into
their discussion, the following will suffice to connect them with the secondary
periodic function.

Let two independent special solutions be g (z) and h (z), uniform functions
of z ; every solution is of the form ag (z} + /3h (z}, where a and /3 are constants.
The equation is unaltered when z + 2w is substituted for z ; hence g {z + 2&>)
and h (z + 2&>) are solutions, so that we must have

g (z + 2w) = Ag (z} + Bh (z}, h(z + 2o>) = Cg (z) + Dh (z\

where, as the functions are determinate, A, B, C, D are determinate constants,
such that AD — BC is different from zero.

Similarly, we obtain equations of the form

g (z + 2co') = A'g (z) + B'h (z\ h(z + 2co') = C'g (z} + D'h (z}.
Using both equations to obtain g (z + 2o> + 2&/) in the same form, we have

BC' = B'C, AB' + BD' = A'B + B'D ;
and similarly, for h (z + 2w + 20)'), we have

C G' A-D A'-U

therefore -~ = -™ = o, — ~ — = — ™ — = e.

x> -D n n

Let a solution F (z} = ag (z) + bh (z)

be chosen, so as to give

if possible. The conditions for the first are

a b

so that a/b (= £) must satisfy the equation

and the conditions for the second are

aA' + bCf aB' + bD'

* Picard, Comptes Rendus, t. xc, (1880), pp. 128—131, 293—295; Crelle, t. xc, (1880), pp.
281—302.

Floquet, Comptes Rendus, t. xcviii, (1884), pp. 82 — 85 ; Ann. de VEc. Norm. Sup., 3mc Ser.,
t. i, (1884), pp. 181—238.

140.] DIFFERENTIAL EQUATION 283

so that £ must satisfy the equation

A'-D'=^B'~~.
These two equations are the same, being

p.-«g-ft*a

Let £j and £2 be the roots of this equation which, in general, are unequal ; and
let fa, fa and fa, fa.' be the corresponding values of /z, //. Then two functions,
say FI (z) and F^ (z), are determined : they are independent of one another, so
therefore are g (z) and h (z) ; and therefore every solution can be expressed in
terms of them. Hence a linear differential equation of the second order, having
coefficients that are doubly-periodic functions of the first kind, can generally be
integrated by means of doubly -periodic functions of the second kind.

It therefore follows that Lame's equation, which will be taken in the form

can be integrated by means of secondary doubly-periodic functions.

141. Let z = c be an accidental singularity of w of order m ; then, for
points z in the immediate vicinity of c, we have

and therefore

2mp

~ z- c + P°SltlVe P°wers °f * -

Since this is equal to n (n + 1) @ (z) + B

it follows that c must be congruent to zero and that m, a positive integer,
must be n. Moreover, p = 0. Hence the accidental singularities of w are
congruent to zero, and each is of order n.

The secondary periodic function, which has no accidental singularities
except those of order n congruent to z = 0, has n irreducible zeros. Let them
be — alt — a2,..., — an; then the form of the function is

Hence 1 *? = ,-»?« +

or, taking p = - ^(ar), we have

and therefore i *? - 1 (*?)' . n(> (,) - X f> (« + «,
19 O^ W2\dzj * v y r»i

284 INTEGRATION

But, by Ex. 3, § 131, we have

[141.

4 r=1 £> (ar) - p (z)

,

by Ex. 4, § 131. Thus

W

Now

r=l S=l

.

g> (a.) -

g> (ar) - g) («) ' g> (a,) - g> (^)

4^?3 (^r) - ^2ip Q) - #, + %>' (a*) & (a,)

where

g> (ar) - £> (a.)
Let the constants a be such that

(O - £> (a2)

+

-H...-0

/i equations of which only n — 1 are independent, because the sum of the n
left-hand sides vanishes. Then iu the double summation the coefficient of

i f .1 f u #>' (ar) — &' (z) .
each of the tractions * )—,- — V\ is zero ; and so

and therefore • -^-, = w (w + 1) p (z) + (2n — 1) 2 ^> (a,.).

/IU GLZ" T=l

Hence it follows that

_<T(z + aJ <T(z + a2)...<r(z + an) -z?J("r)
an (z}

satisfies Lame's equation, provided the n constants a be determined by the
preceding equations and by the relation

B = (2n-l) I pfa).

141.] OF LAMP'S EQUATION 285

Evidently the equation is unaltered when — z is substituted for z ; and
therefore

is another solution. Every solution is of the form

MF(z} + NF(-z),
where M and N are arbitrary constants.

COEOLLARY. The simplest cases are when n = l and n = 2.

When n = 1, the equation is

• j-r- + B :

w dzz

there is only a single constant a determined by the single equation

B = p (a),
and the general solution is

,, a (2 + a) ... ,ra(z — a] ... ,
w = M — ^- 7-^-/ e~2£(a) + N - ---- ' ' es^a>
o- (z} a (z)

When n = 2, the equation is

-J-. = 6(0 (z} + B.

w dz*

The general solution is

^

where a and b are determined by the conditions

Rejecting the solution a+b = 0, we have a and b determined by the equations
p (a)

For a full discussion of Lame's equation and for references to the original sources of
information, see Halphen, Traite des fonctions elliptiques, t. ii, chap, xn., in particular,
pp. 495 et seq.

Ex. When Lamp's equation has the form

1 d?w
- -T-5 =n (n + 1) £2 sn20 - h.

w dz2 ^

obtain the solution for w = l, in terms of the Jacobian Theta-Functions,

where co is determined by the equation dn2o> = A-F ; and discuss in particular the solution
when h has the values l+£2, 1, £2.

Obtain the solution for » = 2 in the form

i +B - fe^)e- K& .1

J SI e(») j'

286 PSEUDO-PERIODIC FUNCTIONS [141.

where X and w are given by the equations

(2P sn2 a - 1 - F) (2F sn2 a - 1) (2 sn2 a - 1)
3Fsn4a-2(l+£2)sn2a + l ~ '

and a is derived from h by the relation

Deduce the three solutions that occur when X is zero, and the two solutions that occur
when X is infinite. (Hermite.)

Doubly-Periodic Functions of the Third Kind.

142. The equations characteristic of a doubly-periodic function <I> (z) of
the third kind are

= <£(», <&(z + 2a)') = e~ »~Z Q(z),

where m is an integer different from zero.

Obviously the number of zeros in a parallelogram is a constant, as well as
the number of infinities. Let a parallelogram, chosen so that its sides
contain no zero and no infinity of <& (z}, have p, p + 2<w, p + 2&>' for three
of its angular points; and let a1} a2, . .., a{ be the zeros and cl5 ..., cm be the
infinities, multiplicity of order being represented by repetitions. Then using

"^ (z) to denote , (log <£ (z)}, we have, as the equations characteristic of

*

and for points in the parallelogram

where -ff (^) has no infinity within the parallelogram. Hence

the integral being taken round the parallelogram : by using the Corollary to
Prop. II. in § 116, we have

27ri (I - n) - - \ - \^L\ dz =
Jp \ &> /

so that I = n + m :

or the algebraical excess of the number of irreducible zeros over the number of

irreducible infinities is equal to in.

z
Again, since — = 1 +

z — /A z — p,

a c

we have 2 2 h I — n = z"^ (z) — zH (z),

z — a z — c

and therefore 2-Tn (Sa — 2c) = jz*\? (z) dz,

142.] OF THE THIRD KIND 287

the integral being taken round the parallelogram. As before, this gives

rp+2<a' rp+2<a < vnTri "I

2™ (2a - 2c) = 2ft)^ (z) dz - MV (z) - - (z + 2ft/) dz.

Jp Jp ( ft) )

The former integral is

rp+*»'(g)

,v x dz
(*)

miri

for the side of the parallelogram contains* no zero and no infinity
The latter integral, with its own sign, is

<P(Z) ft)

= 0 + {O + 2« + 2ft>')2 - (p + 2ft/)2}

= 2TO7T* (p + ft) + 2ft)').

Hence 2a — Sc = m (&) + 2&/),

giving the excess of the sum of the zeros over the sum of the infinities in any
parallelogram chosen so as to contain the variable z and to have no one of its
sides passing through a zero or an infinity of the function.

These will be taken as the irreducible zeros and the irreducible infinities :
all others are congruent with them.

All these results are obtained through the theorem II. of § 116, which
assumes that the argument of <y' is greater than the argument of &) or, what
is the equivalent assumption (§ 129), that

rjco' — w'co = ^iri.

143. Taking the function, naturally suggested for the present class by
the corresponding function for the former class, we introduce a function

a(z- d) <r(z- C2). ..<r(z — Cn) '

where the a's and the c's are connected by the relations
Sa — Sc = m (&) + 2&>'), l—n = m.

Then (f>(z) satisfies the equations characteristic of doubly-periodic functions
of the third kind, if

0 = 4Xo) + 2ra77,
k . 27rt = 4X&)2 + 2m?/ft) + 2/ift) + miri — Zmrj (&> + 2ft)') ;

miri — 2mrj' (&> + 2ft)'),

* Both in this integral and in the next, which contain parts of the form I — there is, as in

J w

Prop. VII., § 116, properly an additive term of the form 2iciri, where K is an integer ; but, as there,
both terms can be removed by modification of the position of the parallelogram, and this modifi
cation is supposed, in the proof, to have been made.

288 TERTIARY FUNCTIONS [143.

k and k' being disposable integers. These are uniquely satisfied by taking

with A; = 0, k' = m.

Assuming the last two, the values of X and /JL are thus obtained so as to make
<fr (z) a doubly-periodic function of the third kind.

Now let Oj, ..., di be chosen as the irreducible zeros of <l> (z) and Ci, ..., cn
as the irreducible infinities of <E> (2), which is possible owing to the conditions
to which they were subjected. Then <3> (z)/<j> (z) is a doubly-periodic function
of the first kind; it has no zeros and no infinities in the parallelogram of-
periods and therefore none in the whole plane ; it is therefore a constant, so
that

3> (z) = Ae"** "IZ*+^ - + (l|+8'')} ** <r(*-gi)°-(*-q»)-. •*(*-<*!)

tr(z- d) <r(z- c.2)...o- (z - cn) '

a representation of <3> (z) in terms of known quantities.

Ex. Had the representation been effected by means of the Jacobian Theta-Functions
which would replace a (z) by H(z), then the term in z1 in the exponential would be absent.

144. No limitation on the integral value of m, except that it must not
vanish, has been made : and the form just obtained holds for all values.
Equivalent expressions in the form of sums of functions can be constructed :
but there is then a difference between the cases of m positive and m
negative.

If m be positive, being the excess of the number of irreducible zeros over
the number of irreducible infinities, the function is said to be of positive class
m ; it is evident that there are suitable functions without any irreducible
infinities — they are integral functions.

When m is negative (= — n), the function is said to be of negative class n ;
but there are no corresponding integral functions.

145. First, let m be positive.

i. If the function have no accidental singularities, it can be expressed in
the form

A e**+i* a-(z — a1)a-(z — aa)...<r(z — am),

with appropriate values of X and //..

ii. If the function have n irreducible accidental singularities, then it has
m + n irreducible zeros. We proceed to shew that the function can be
expressed by means of similar functions of positive class m, with a single
accidental singularity.

145.] OF POSITIVE CLASS 289

Using X and /j, to denote

, mri

- 1 — ' and | - - + m (77 + 277'),
&) a)

which are the constants in the exponential factor common to all functions of
the same class, consider a function, of positive class m with a single accidental
singularity, in the form

*m (z, u) = eW ''

<r(u- 6X) o- (u - &„). • • <r (u - bm+1) <r(z-u)'
where b1} b.2, ..., bm are arbitrary constants, of sum s, and
m (&> + 2ft)') = 6OT+1 + fcj + b.> + . . . bm - u

= bm+l +s-u.
The function y-m satisfies the equations

_mirzi

y-w (z + 2<w, u) = i/rm (z, u), y,tt (z + 2&)', w) = e~' « -^m (z, u) ;

regarded as a function of z, it has u for its sole accidental singularity,
evidently simple.

The function - — can be expressed in the form

I\I/* I £ It I

u — k) . . . a- (u — bm) o- {s — m (&)

(r^-b,) ............ a-(z-bm) a{u- z-s + m(a> + 2~w7)} '

Regarded as a function of u, it has z, \, . . ., bm for zeros and z + s - m (to + 2o>')
for its sole accidental singularity, evidently simple : also

z + &J + ...+ bm - {z + s - m (&) + 2o/)} = m (w + 2o>').

Hence owing to the values of X arid p, it follows that -- } - x when re-

f>m(*, tt)

garded as a function of u, satisfies all the conditions that establish a doubly-
periodic function of the third kind of positive class m, so that

1 1

~i 7 ~ =r ^

and therefore

mnz

tym (z, u + 2o>) = ^m (z, u), ^m (z, u + 20)') = e~ijrm (z, u).
Evidently -f m (z, u) regarded as a function of u is of negative class m : its
infinities and its sole zero can at once be seen from the form

-bm) o-{u-z-s+m(ca

<r(u -z)*^-^)...*^- bm) a- {s - m (to + 2o)')j '

Each of the infinities is simple. In the vicinity of u = z, the expansion of
the function is

^^z + positive integral powers of u — z :

19

290 TERTIARY FUNCTIONS [145.

and, in the vicinity of u = br, it is

C* ( 7\

r j + positive integral powers of u — br,

Lv "~~ \Jrp

where Gr (z) denotes

r) <r(z-bi)--.<r(z-br-i)<r(z-br+l)...a(z-bm)o-{z+s-br-m(a>+2a>')}
a- (br - 6j). ..cr (br - 6r_!) cr(br - br+l)...cr(br - bin) o-[s-ra(eo + 2&>')}'

and is therefore an integral function of z of positive class m.

Let 4> (14) be a doubly-periodic function of the third kind, of positive class
m ; and let its irreducible accidental singularities, that is, those which occur
in a parallelogram containing the point u, be a^ of order !+/*!, a., of order
1 + ju,2, and so on. In the immediate vicinity of a point ar, let

--... ±

\

-

rr— r;r-... r-,-~- - -rr.

cm du- du^J u — a,.

Then proceeding as in the case of the secondary doubly-periodic functions
(§ 137), we construct a function

F(u) = 3?(u)^m(z, u).
We at once have F (u + 2o>) = F (u) = F(u + 2a>'),

so that F(u) is a doubly-periodic function of the first kind; hence the sum
of its residues for all the poles in a parallelogram of periods is zero.

For the infinities of F (u), which arise through the factor tym(z, u}, wea
have as the residue for u = z

-<*>(*),
and as the residue for u = br, where r = 1, 2, ..., m,

In the vicinity of a,., we have

fyn (Z, u) = ^rm (Z, «r) + (u - Or) tym' (z, O.r)

where dashes imply differentiation of ^rm {z, u} with regard to u, after which
u is made equal to a,. ; so that in <I> (u) tym (z, u) the residue for u = ar, where
r = l, 2, ..., is

Er (z) = Ar ,jrm (z, ctr) + B, Tjrm' (z, a,.) + Cr tym" (z, ar) + ...+ Mr <^m^r) (z> ar\
Hence we have

and therefore ®(z)= 2 E,(z)+ 2 <& (br) Gr(z),

s=l r=l

giving the expression of <l> (z) by means of doubly -periodic functions of tht
third kind, which are of positive class m and have either no accidental singu->
larity or only one and that a simple singularity.

145.] OF NEGATIVE CLASS 291

The m quantities blt ..., bm are arbitrary; the simplest case which occurs
is when the m zeros of &(z) are different and are chosen as the values
of &!,..., bm. The value of 3>(z) is then

<&(*)= 2 JS'.C*),

s=l

where the summation extends to all the irreducible accidental singularities ;
while, if there be the further simplification that all the accidental singularities
are simple, then

<I> (z) = A1 TJrm (2, «!> + As tym (z, ot2) + . . .,

the summation extending to all the irreducible simple singularities.
The quantity tym (z, ar), which is equal to

) <r(z-bd...<r(z- bm) <r{z + 2b-m(<o + 2ft/) - ar]

a-(ar — b1)...a- (ar - bm) <r {26 - m (co + 2ft>')} a- (z - ar) '

and is subsidiary to the construction of the function E (z\ is called the
simple element of positive class m.

In the general case, the portion

gives an integral function of z, and the portion 2 Es (z) gives a fractional

s=l

function of z.

146. Secondly, let m be negative and equal to — n. The equations
satisfied by & (z} are

i = <I> 0), <I> (z + 2ft)') = e w <£ 0),

and the number of irreducible singularities is greater by n than the number
of irreducible zeros.

One expression for <i> (z} is at once obtained by forming its reciprocal,
which satisfies the equations

11 1 -2-** i

f\ /K / -\ >

and is therefore of the class just considered: the value of is of the

q>(^)

form

ZEs(z) + ^ArGr(z}.

For purposes of expansion, however, this is not a convenient form as it gives
only the reciprocal of <I> (z}.

To represent the function, Appell constructed the element

TT sv°° Ffr-K»-*Wl 7r(2
gr— * . • cot — *-

19—2

292 TERTIARY FUNCTIONS [146.

which, since the real part of to' fan is positive, converges for all values of z and

y, except those for which

z = y (mod. 2&>, 2&>').

For each of these values one term of the series, and therefore the series
itself, becomes infinite of the first order.

Evidently %„ (z, y + 2o>) = %M (z, y},

niryi

Xn (z, y + 2eo') = e ° %„(*, y);
therefore in the present case

0(y)=*(3f)jfr (**?)>

regarded as a function of ^/, is a doubly-periodic function of the first kind.

Hence the sum of the residues of its irreducible accidental singularities
is zero.

When the parallelogram is chosen, which includes z, these singularities
are

(i) y = z, arising through %n (z, y} ;

(ii) the singularities of <£ (y}, which are at least n in number, and are
n + I when <& has I irreducible zeros.

The expansion of Xn 0> y), in powers of y - z, in the vicinity of the point
z, is

+ positive integral powers of y — z ;

y-z
therefore the residue of II (y) is

Let ctr be any irreducible singularity, and in the vicinity of a,, let 3> (y) denote

d

-I- positive integral powers of y — Or,

where the series of negative powers is finite because the singularity is
accidental ; then the residue of H (y} is

Ar ^ (Z, Or) + Br Xn (*, «r) + Cr %,/' (z, Ct,) + . . . + Pr X*™ 0> «>')>

where %n(A) (^, ar) is the value of

dx%n (z, y)

dy*

when y = 0r after differentiation. Similarly for the residues of other singu
larities : and so, as their sum is zero, we have

<£ (Z) = 2 {Ar Xn (*, «r) + Br Xn (*, «•,) + ...+ P, XnW (?, «r)},

the summation extending over all the singularities.

146.] OF NEGATIVE CLASS 293

The simplest case occurs when all the N(>n) singularities a are accidental
and of the first order ; the function 4> (z) can then be expressed in the form

Al Xn (Z, «i) + A2 Xn (Z, Oj) + . . . + AN Xn (z, «#)•

The quantity Xn (z, a), which is equal to

T *^" ^p{(«-i)»'+«} TT 0 - a

a 2/6 COt — -^.

.

2(0

is called the simple element for the expression of a doubly-periodic function of
the third kind of negative class n.

Ex. Deduce the result

_ ^ ( — iVcot
TT snu s=-oov I 2K /'

147. The function Xn (z, y} can be used also as follows. Since Xm (z, y),
qua function of y, satisfies the equations

%m (z, 11 + 2(i)} = Y™ (z, 7/\
llv \ s {/ ' / /V//fc \ J ts /'

miryi

Xm (z, y + 2o/) = e~^xm (z, y),

which are the same equations as are satisfied by a function of y of positive
class m, therefore Xm (<*> z), which is equal to

2 e cot

being a function of z, satisfies the characteristic equations of § 142 ; and, in
the vicinity of z = a,

Xm (a> z) — - — + positive integral powers of z — a.

Z ~~" OC

If then we take the function 4> (z) of § 145, in the case when it has simple
singularities at alt «2, ... and is of positive class m, then

4> (z) + A, xw (a, ,

is a function of positive class m without any singularities: it is therefore
equal to an integral function of positive class m, say to G(z)t where

G (z) = Ae^+^a- (z-al}...(r(z- am),
so that 3>(z) = G(z)-A1Xm(ct1,2)-A,xm(<Xt,z)-....

Ex. As a single example, consider a function of negative class 2, and let it have no
zero within the parallelogram of reference. Then for the function, in the canonical
product-form of § 143, the two irreducible infinities are subject to the relation

and the function is * (z) = AV° "V" -

o- (z— Cj) o- (z-c2)'

294 TERTIARY FUNCTIONS [147.

The simple elements to express 3> (z) as a sum are

2.<!iri , ,

» {{s-lX + Cl} ,77, ' «

*« " «rt (s-C! -2*,),

4iri, ,,
7T -(ci-<o)» - r-w-c'i TT

= _e<-> 2 e a> cot — (2 + 0 j-2no)

after an easy reduction,

4irj

The residue of *(s) for cn which is a simple singularity, is

'Us-(
Al = Ktfa v<

and for c2, also a simple singularity, it is

,

so that ^- = -ew =-ew

^2

Hence the expression for 4> (z) as a sum, which is

!

becomes Al (X2 (2, Cj) - e u ^2 (^ - ci)}

that is, it is a constant multiple of

Again,

— j - -

<r(z- GJ) a- (z + c^ - 2o>-

on changing the constant factor. Hence it is possible to determine L so that

•ni Tti

" C' « c - e<a

Taking the residues of the two sides for z=c1} we have
and therefore finally we have

-C]*- — Ci -- C,

Le™ <» = e °>

>-.•>-*

TtlC

(a (s, c) - e w X2 (2> - c)

<* Cot^L(2-c1-2su)')-e w cot - - (z + cx - 2sw') K

2<a 2a> )

the right-hand side of which admits of further modification if desired.

147.] PSEUDO-PERIODIC FUNCTIONS 295

Many examples of such developments in trigonometrical series are given by Hermite*,
Biehlerf, HalphenJ, Appell§, and Krause||.

148. We shall not further develop the theory of these uniform doubly-
periodic functions of the third kind. It will be found in the memoirs of
Appell§ to whom it is largely due; and in the treatises of Halphen**, and
of Rausenberger"f"f.

It need hardly be remarked that the classes of uniform functions of a
single variable which have been discussed form only a small proportion of
functions reproducing themselves save as to a factor when the variable
is subjected to homographic substitutions, of which a very special example
is furnished by linear additive periodicity. Thus there are the various
classes of pseudo-automorphic functions, (§ 305) called Thetafuchsian by Pom-
care, their characteristic equation being

for all the substitutions of the group determining the function : and other
classes are investigated in the treatises which have just been quoted.

The following examples relate to particular classes of pseudo-periodic
functions.

Ex. 1. Shew that, if F (z) be a uniform function satisfying the equations

m

where b is a primitive mth root of unity, then F(z) can be expressed in the form

where f(z) denotes the function

and prove that \F(z)dz can be expressed in the form of a doubly-periodic function
together with a sum of logarithms of doubly-periodic functions with constant coefficients.

(Goursat.)

* Comptes Rendus, t. Iv, (1862), pp. 11—18.

t Sur les developpements en series des fonctions doublement periodiqucs de troisieme espece,
(These, Paris, Gauthier-Villars, 1879).

£ Traite des fonctions elliptiques, t. i, chap. xm.

§ Annales de VEc. Norm. Sup., 3rae S6r., t. i, pp. 135—164, t. ii, pp. 9—36, t. iii, pp. 9—42.

|| Math. Ann., t. xxx, (1887), pp. 425—436, 516—534.

'* Traite des fonctions elliptiques, t. i, chap. xiv.

ft Lehrbuch der Theorie der periodischen Functional, (Leipzig, Teubner, 1884), where further
references are given.

296 PSEUDO-PERIODIC FUNCTIONS [148.

Ex. 2. Shew that, if a pseudo-periodic function be denned by the equations

and if, in the parallelogram of periods containing the point z, it have infinities c, ... such
that in their immediate vicinity

then/ (2) can be expressed in the form

-'^'^«{^I+ ...... +«»,£}«—>,

the summation extending over all the infinities of/ (z) in the above parallelogram of periods,
and the constants (715 ... being subject to the condition

+ iVS Cl = A o>' — X'«o.

Deduce an expression for a doubly-periodic function <f) (z) of the third kind, by
assuming

/W-f]8. (Halphen.)

(f> \g)

Ex. 3. If S(z) be a given doubly-periodic function of the first kind, then a
pseudo-periodic function F(z), which satisfies the equations

F(z + ^} = F(z),
mriz
F (z + 2o>') = e ~"~ S (z} F (z),

where n is an integer, can be expressed in the form

where -4 is a constant and TT (2) denotes

the summation extending over all points &,. and the constants Br being subject to the
relation

Explain how the constants b, G and B can be determined. (Picard.)

Ex. 4. Shew that the function F(z) defined by the equation

for values of \z\, which are <1, satisfies the equation

and that the function Fl(a!)=^ ^rjr-£i

where (j)(,v) = 3? — 1, and </>„(.*•')> f()r positive and negative values of n, denotes (/> [0 {<£ 0 (#)}]

<f> being repeated n times, and a is the positive root of a3 — a - 1 = 0 ; satisfies the equation

for real values of the variable.

Discuss the convergence of the series which defines the function Fl (x). (Appell.)

CHAPTER XIII.

FUNCTIONS POSSESSING AN ALGEBRAICAL ADDITION-THEOREM.

149. WE may consider at this stage an interesting set* of important
theorems, due to Weierstrass, which are a justification, if any be necessary,
for the attention ordinarily (and naturally) paid to functions belonging to
the three simplest classes of algebraic, simply-periodic and doubly-periodic
functions.

A function <f> (u) is said to possess an algebraical addition theorem, when
among the three values of the function for arguments u, v, and u + v, where u
and v are general and not merely special arguments, an algebraical equation
exists f having its coefficients independent of u and v.

150. It is easy to see, from one or two examples, that the function does
not need to be a uniform function of the argument. The possibility of
multiformity is established in the following proposition :

A function defined by an algebraical equation, the coefficients of which are
uniform algebraical functions of the argument, or are uniform simply -periodic
functions of the argument, or are uniform doubly -periodic functions of the
argument, possesses an algebraical addition-theorem.

* They are placed in the forefront of Schwarz's account of Weierstrass's theory of elliptic
functions, as contained in the Formeln und Lehrsdtze zum Gebrauche der elliptischen Functionen;
but they are there stated (§§ 1—3) without proof. The only proof that has appeared is in a
memoir by Phragmen, Acta Math., t. vii, (1885), pp. 33—42; and there are some statements
(pp. 390—393) in Biermann's Theorie der analytischen Functionen relative to the theorems. The
proof adopted in the text does not coincide with that given by Phragme'n.

t There are functions which possess a kind of algebraical addition -theorem ; thus, for
instance, the Jacobian Theta-functions are such that eA(u + w) O^ (u- v) can be rationally ex
pressed in terms of the Theta-functions having it and v for their arguments. Such functions
are, however, naturally excluded from the class of functions indicated in the definition.

Such functions, however, possess what may be called a multiplication-theorem for multipli
cation of the argument by an integer, that is, the set of functions 6 (nut) can be expressed
algebraically in terms of the set of functions 6 (M). This is an extremely special case of a set
of transcendental functions having a multiplication-theorem, which are investigated by Poincare,
Liouville, 4°" S6r., t. iv, (1890), pp. 313—365.

298 EXAMPLES OF FUNCTIONS [150.

First, let the coefficients be algebraical functions of the argument u. If
the function defined by the equation be U, we have

Umg0 (u) + Um~lgi (u) + ...+gm (u) = 0,

where g0(u),gi(u}, ...,gm(u) are rational integral algebraical functions of u
of degree, say, not higher than n. The equation can be transformed into

un f/U\+ u'1-1/! ( U) + ... + fn ( U) = 0,

where f0(U), fi(U), ••••> fn(U) are rational integral algebraical functions of
U of degree not higher than m.

If V denote the function when the argument is v, and W denote it when
the argument is u + v, then

w»/0 (7) + ^1/1 (7) + ... +fn (V) M 0,
and (u + v)n/0 ( W) + (u + vY^f, ( W ) + . . . +fn ( W ) = 0.

The algebraical elimination of the two quantities u and v between these
three equations leads to an algebraical equation between the quantities
/(£/"), /(7) and f (W), that is, to an algebraical equation between U, V, W,
say of the form

G(U, V, F) = 0,

where G denotes an algebraical function, with coefficients independent of
u and v. It is easy to prove that G is symmetrical in U and 7, and that
its degree in each of the three quantities U, 7, W is wn2. The equation
G = 0 implies that the function U possesses an algebraical addition- theorem.

Secondly, let the coefficients* be uniform simply-periodic functions of
the argument u. Let &> denote the period: then, by § 113, each of these

TT'IL

functions is a rational algebraical function of tan — . Let u' denote

tan — ; then the equation is of the form

Umg0 (u') + Um^g, (u'} + ...+ gm 00 = 0,

where the coefficients g are rational algebraical (and can be taken as
integral) functions of u'. If p be the highest degree of u' in any of them,
then the equation can be transformed into

u'vfo ( U) + u'P-1/! ( U) + . . . + fp ( U) = 0,

where f0(U), fi(U), ..., fp(U) are rational integral algebraical functions of
U of degree not higher than m.

* The limitation to uniformity for the coefficients has been introduced merely to make the
illustration simpler; if in any case they were multiform, the equation would be replaced by
another which is equivalent to all possible forms of the first arising through the (finite)
multiformity of the coefficients : and the new equation would conform to the specified
conditions.

150.] POSSESSING AN ADDITION-THEOREM 299

Let v denote tan — , and w denote tan — -- ; then the corresponding
cy &)

values of the function are determined by the equations

and w'*>f0(W) + w'p-*/! (W) + ... +fp (W) = 0.

The relation between u', v', w' is

u'v'w' + u' + v' - w' = 0.

The elimination of the three quantities u', v', w' among the four equations
leads as before to an algebraical equation

G(U, V, W) = 0,

where G denotes an algebraical function (now of degree mp'2) with coefficients
independent of u and v. The function U therefore possesses an algebraical
addition-theorem.

Thirdly, let the coefficients be uniform doubly-periodic functions of the
argument u. Let &> and &/ be the two periods ; and let @ (u), the Weier-
strassian elliptic function in those periods, be denoted by £. Then every
coefficient can be expressed in the form

~L '

where L, M, N are rational integral algebraical functions of f of finite
degree. Unless each of the quantities N is zero, the form of the equation
when these values are substituted for the coefficients is

A+Bp'(u) = 0,

so that A* = &(±?-g£-9*)\

and this is of the form

Umff* (£) + U'^g, (|) + . . . + gm (£) - 0,

where the coefficients g are rational algebraical (and can be taken as integral)
functions of £ If q be the highest degree of £ in any of them, the equation
can be transformed into

where the coefficients / are rational integral algebraical functions of U of
degree not higher than 2m.

Let TJ denote $ (v) and f denote p(u + v); then the corresponding values
of the function are determined by the equations

......... +fq(V)=0,

By using Ex. 4, § 131, it is easy to shew that the relation between £, rj, £ is

300 WEIERSTRASS'S THEOREM ON FUNCTIONS [150.

The elimination of £, ij, £ from the three equations leads as before to an

algebraical equation

G(U,V, W) = 0,

of finite degree and with coefficients independent of u and v. Therefore in this
case also the function U possesses an algebraical addition-theorem.

If, however, all the quantities N be zero, the equation defining U is of the

form

Umh0 (£) + U^h, (£) + . . . + hm (£) = 0,

and a similar argument then leads to the inference that U possesses an
algebraical addition-theorem.

The proposition is thus completely established.

151. The generalised converse of the preceding proposition now suggests
itself : what are the classes of functions of one variable that possess an alge
braical addition-theorem? The solution is contained in Weierstrass's theorem : —

An analytical function <f> (u), which possesses an algebraical theorem, is
either

(i) an algebraical function of u ; or

liru

(ii) an algebraical function of e » , where w is a suitably chosen
constant ; or

(iii) an algebraical function of the elliptic function %>(u), the periods — or
the invariants g.z and g3 — being suitably chosen constants.

Let U denote </> (w).

For a given general value of u, the function U may have m values where,
for functions in general, there is not a necessary limit to the value of m ; it
will be proved that, when the function possesses an algebraical addition-
theorem, the integer m must be finite.

For a given general value of U, that is, a value of U when its argument is
not in the immediate vicinity of a branch-point if there be branch-points, the
variable u may have p values, where p may be finite or may be infinite.

Similarly for given general values of v and of V, which will be used to
denote <£ (v).

First, let p be finite. Then because u has p values for a given value of U
and v has p values for a given value of V, and since neither set is affected by the
value of the other function, the sum u + v has p2 values because any member of
the set u can be combined with any member of the set v ; and this number
p2 of values of u + v is derived for a given value of U and a given value of V.

Now in forming the function <j>(u + v), which will be denoted by W, we
have m values of W for each value of u + v and therefore we have mp2 values
of W for the whole set, that is, for a given value of U and a given value of V.

151.] POSSESSING AN ADDITION-THEOREM 301

Hence the equation between U, V, W is of degree* mp2 in W, necessarily
finite when the equation is algebraical ; and therefore m is finite.

Because m is finite, U has a finite number m of values for a given value of
u ; and, because p is finite, u has a finite number p of values for a given value of
U. Hence U is determined in terms of u by an algebraical equation of degree
m, the coefficients of which, are rational integral algebraical functions of
degree p ; and therefore U is an algebraic function of u.

152. Next, let p be infinite ; then (see Note, p. 303) the system of values
may be composed of (i) a single simply-infinite series of values or (ii) a finite
number of simply-infinite series of values or (iii) a simply-infinite number of
simply-infinite series of values, say, a single doubly-infinite series of values or
(iv) a finite number of doubly-infinite series of values or (v) an infinite
number of doubly-infinite series of values where, in (v), the infinite number
is not restricted to be simply-infinite.

Taking these alternatives in order, we first consider the case where the p
values of u for a given general value of U constitute a single simply -infinite
series. They may be denoted by f (u, n), where n has a simply-infinite
series of values and the form of/ is such that f(u, 0) = u.

Similarly, the p values of v for a given general value of V may be denoted
by/(y, n), where n' has a simply-infinite series of values. Then the different
values of the argument for the function W are the set of values given by

f(u,n)+f(v,ri),

for the simply-infinite series of values for n and the similar series of values
for n'.

The values thus obtained as arguments of W must all be contained in
the series f(u + v, n"}, where n" has a simply-infinite series of values ; and,
in the present case,/(w + w, n"} cannot contain other values. Hence for some
values of n and some values of n', the total aggregate being not finite, the
equation

f(u,n}+f(v,n'}=f(u + v,n")
must hold, for continuously varying values of u and v.

In the first place, an interchange of u and v is equivalent to an interchange
of n and n on the left-hand side; hence n" is symmetrical in n and n'.
Again, we have

df(u, n) _ df(u + v, n")
du 3 (u + v)

dv '

* The degree for special functions may be reduced, as in Cor. 1, Prop. XIII, § 118; but in no
case is it increased. Similarly modifications, in the way of finite reductions, may occur in the
succeeding cases ; but they will not be noticed, as they do not give rise to essential modification
in the reasoning.

302 FORM OF ARGUMENT [152.

so that the form of f(u, n) is such that its first derivative with regard to u is
independent of u. Let 0 (n) be this value, where 0 (n), independent of u, may
be dependent on n ; then, since

we have f(u, n) = uO (n) + ty (n),

-fy- (n) being independent of u. Substituting this expression in the former

equation, we have the equation

u6 (n) + ^ (n) + v9 (n'} + f (71') = (u + v)6 (n"} + ^ (n"),
which must be true for all values of u and v ; hence

e(n)=e(n") = d(n'),

so that 6 (n) is a constant and equal to its value when n = 0. But when n is
zero,/(w, 0) is u ; so that 9 (0) = 1 and ^ (0) = 0, and therefore

f(u, n) = u + Tjr (n),
where i/r vanishes with n.

The equation defining ty is

for values of n from a singly-infinite series and for values of n' from the same
series, that series is reproduced for TO". Since ^ (n) vanishes with n, we take

^ (n) = HX (n),

and therefore rc% (n) + n'% (n') = ri'x (n").

Again, when n' vanishes, the required series of values of n" is given by taking
n" = n ; and, when n does not vanish, n" is symmetrical in n and n', so that

we have

n" = n + n' + nn\,

where X is not infinite for zero or finite values of n or n'. Thus
•HX (n) + n'x (n) = (n + TO' + -nw'X) % (w + ?*' + wi'X).

Since the left-hand side is the sum of two functions of distinct and inde
pendent magnitudes, the form of the equation shews that it can be satisfied

only if

X = 0, so that n" = n + n' ;

and % 0) = % (n//)

= %(n'\
so that each is a constant, say o> ; then

f(u, n} = u + nco,

which is the form that the series must adopt when the series f(u + v, n") is
obtained by the addition of/(«, n) and/0, n')-

152.] IN A SIMPLY-INFINITE SERIES 303

It follows at once that the single series of arguments for W is obtained,
as one simply-infinite series, of the form u + v+n"a). For each of these
arguments we have m values of W, and the set of m values of W is
the same for all the different arguments; that is, W has m values for a
given value of U and a given value of V. Moreover, U has m values for each
argument and likewise V; hence, as the equation between U, V, W is of
a degree that is necessarily finite because the equation is algebraical, the
integer m is finite.

It thus appears that the function U has a finite number m of values for
each value of the argument u, and that for a given value of the function the
values of the argument form a simply-periodic series represented by u + nw.

But the function tan ( — ) is such that, for a given value, the values of the

V 03 J

argument are represented by the series u + nw ; hence for each value of

tan ( — 1 there are m values of U and for each value of U there is one value
\ «o /

of tan -- . It therefore follows, by SS 113, 114, that between U and tan (— }
w \ to /

there is an algebraical relation which is of the first degree in tan - - and the

O)

U

rath degree in U, that is, U is an algebraic function of tan — - . Hence U is

(I)

an algebraic function also of e <" .

Note. This result is based upon the supposition that the series of argu
ments, for which a branch of the function has the same value, can be arranged
in the form/(w, n), where n has a simply-infinite series of integral values. If,
however, there were no possible law of this kind — the foregoing proof shews
that, if there be one such law, there is only one such law, with a properly
determined constant co — then the values would be represented by ul} u», ...,up
with p infinite in the limit. In that case, there would be an infinite number of
sets of values for u + v of the type WA + v^, where X and p might be the same
or might be different ; each set would give a branch of the function W and then
there would be an infinite number of values of W corresponding to one branch
of U and one branch of V. The equation between U, V and W would be of
infinite degree in W, that is, it would be transcendental and not algebraical.
The case is excluded by the hypothesis that the addition-theorem is alge
braical, and therefore the equation between U, V and W is algebraical.

153. Next, let there be a number of simply-infinite series of values of
the argument of the function, say q, where q is greater than unity and
may be either finite or infinite. Let ul} u.2, ..., uq denote typical members
of each series.

Then all the members of the series containing ul must be of the form

304 FORM OF ARGUMENT [153.

fi (ui> n)> f°r an infinite series of values of the integer n. Otherwise, as in the
preceding note, the sum of the values in the series of arguments u and of
those in the same series of arguments v would lead to an infinite number of
distinct series of values of the argument u + v, with a corresponding infinite
number of values W ; and the relation between U, V, W would cease to be
algebraical.

In the same way, the members of the corresponding series containing ^
must be of the form/! (v1} ri) for an infinite series of values of the integer n'.
Among the combinations

the simply-infinite series fi(tii+v1} n") must occur for an infinite series
of values of n"; and therefore, as in the preceding case,

fi(uly n) = M1 + nw1,

where toj is an appropriate constant. Further, there is only one series of
values for the combination of these two series ; it is represented by

Ui + v1 + n"wl.

In the same way, the members of the series containing u2 can be repre
sented in the form u2 + nco2, where o>2 is an appropriate constant, which may
be (but is not necessarily) the same as Wj ; and the series containing u.2,
when combined with the set containing v2, leads to only a single series
represented in the form u.2 + v2 + ri'o)2. And so on, for all the series in order.

But now since u2 + m2a)2, where m2 is an integer, is a value of u for a given
value of U, it follows that U (u2 + ra2a>2) = U (w2) identically, each being equal

to U. Hence

U (M! + mlwl + 7n.,<y2) = U (i^ + ra^) = U (u^ = U,

and therefore ^ + ml(al + ra2&>2 is also a value of u for the given value of U,
leading to a series of arguments which must be included among the original
series or be distributed through them. Similarly u1 + 2mr(i)r, where the
coefficients ra are integers and the constants to are properly determined,
represents a series of values of the variable u, included among the original
series or distributed through them. And generally, when account is taken of
all the distinct series thus obtained, the aggregate of values of the variable u
can be represented in the form Wx+2wrtur, for \ — 1, 2, ..., K, where K is
some finite or infinite integer.

Three cases arise, (a) when the quantities « are equal to one another or
can be expressed as integral multiples of only one quantity a>, (6) when the
quantities &> are equivalent to two quantities f^ and O2 (the ratio of which is
not real), so that each quantity &> can be expressed in the form

a>r=plrfil+parsia>

the coefficients plr, p2r being finite integers ; (c) when the quantities « are
not equivalent to only two quantities, such as flj and fl2.

153.] SIMPLY-PERIODIC FUNCTIONS 305

For case (a), each of the K infinite series of values u can be expressed
in the form u^+pci), for X = 1, 2, ..., « and integral values of p.

First, let K be finite, so that the original integer q is finite. Then the
values of the argument for W are of the type

that is, MA + '?V +£>"&>,

for all combinations of \ and fju and for integral values of p". There are thus
K- series of values, each series containing a simply-infinite number of terms
of this type.

For each of the arguments in any one of these infinite series, W has ra
values ; and the set of m values is the same for all the arguments in one and
the same infinite series. Hence W has w/c2 values for all the arguments in
all the series taken together, that is, for a given value of U and a given
value of V. The relation between U, V, W is therefore of degree m«2,
necessarily finite when the equation is algebraical ; hence m is finite.

It thus appears that the function U has a finite number m of values for
each value of the argument u, and that for a given value of the function there
are a finite number K of distinct series of values of the argument of the form

7TU

u+poi), w being the same for all the series. But the function tan -- has
one value for each value of u and the series u+pat represents the series of

7TU

values of u for a given value of tan — . It therefore follows that there are

CO

m values of U for each value of tan — and that there are K values of tan —

to o>

for each value of U ; and therefore there is an algebraical relation between

U and tan — , which is of degree K in the latter and of degree m in the
&)

iiru
TTlI

former. Hence U is an algebraic function of tan — and therefore also of e M .

Next, let K be infinite, so that the original integer q is infinite. Then,
as in the Note in § 152, the equation between U, V, W will cease to be
algebraical unless each aggregate of values u^+pw, for each particular
value of p and for the infinite sequence X= 1, 2, ..., K, can be arranged in a
system or a set of systems, say a in number, each of the form fp(u+pa), pp)
for an infinite series of values of pp. Each of these implies a series of values
fp(v+p'u>, pp) of the argument of V for the same series of values of pp as of
pp> and also a series of values fp(u + v+p"(o, pp") of the argument of W for
the same series of values of pp". By proceeding as in § 152, it follows that

fp (u +pa>, pp} = u+pto +pp(0p,

where &>p' is an appropriate constant, the ratio of which to &> can be proved
F. 20

306 FORM OF ARGUMENT [153.

(as in § 106) to be not purely real, and pp has a simply-infinite succession of
values. The integer a may be finite or it may be infinite.

When ay and all the constants o>' which thus arise are linearly equivalent
to two quantities f^ and O2, so that the terms additive to u can be expressed
in the form 8^ + s.2fl», then the aggregate of values u can be expressed
in the form

for a simply-infinite series for pl and for p2 ; and p has a series of values
1, 2, ..., <r. This case is, in effect, the same as case (6).

When o) and all the constants «' are not linearly equivalent to only
two quantities, such as Oj and IL>, we have a case which, in effect, is the
same as case (c).

These two cases must therefore now be considered.

For case (6), either as originally obtained or as derived through parfc
of case (a), each of the (doubly) infinite series of values of u can be expressed
in the form

for X = 1, 2, ..., <r and for integral values of _p, and p,. The integer a may be
finite or infinite ; the original integer q is infinite.

First, let cr be finite. Then the values of the argument for W are of the
type

that is, u\ + v^ +pi"£li + p2"O2,

for all combinations of \ and p and for integral values of £>/' and p.". There
are thus cr2 series of values, each series containing a doubly-infinite number ofl
terms of this type.

For every argument there are m values of W ; and the set of m values is
the same for all the arguments in one and the same infinite series. Thus W
has mo-2 values for all the arguments in all the series, that is, for a given value
of U and a given value of V; and it follows, as before, from the consideration i
of the algebraical relation, that m is finite.

The function U thus has m values for each value of the argument u ; and
for a given value of the function there are cr series of values of the argument,
each series being of the form wx + PI^I +p.2Q*-

Take a doubly-periodic function © having Oj and H2 for its periods, such*1
that for a given value of © the values of its arguments are of the foregoing
form. Whatever be the expression of the function, it is of the order cr. ,
Then U has m values for each value of @, and @ has one value for each'.
value of U; hence there is an algebraical equation between U and ©, ow

* All that is necessary for this purpose is to construct, by the use of Prop. XII, § 118, ai
function having, as its irreducible simple infinities, a series of points aj, a2,..., a<7 — special*
values of «j, w2, ..., ua— in the parallelogram of periods, chosen so that no two of the <r points a
coincide.

153.] DOUBLY-PERIODIC FUNCTIONS 307

:he first degree in the latter and of the rath degree in U: that is, U is an
algebraical function of @. But, by Prop. XV. § 119, © can be expressed in
the form

where L, M, N are rational integral algebraical functions of $ (u), if f^ and H2
be the periods of g) (u); and g)' (u) is a two- valued algebraical function of jjp (u),
so that © is an algebraical function of i@ (u). Hence also U is an algebraical
function of $(u\ the periods o/<p (u) being properly chosen.

This inference requires that a, the order of ©, be greater than 1.
Because U has m values for an argument u, the symmetric function St/"
has one value for an argument u and it is therefore a uniform function.
But each term of the sum has the same value for u+pifli+pflt as for
u ; and therefore this uniform function is doubly-periodic. The number of
independent doubly-infinite series of values of u for a uniform doubly-
periodic function is at least two : and therefore there must be at least two
doubly-infinite series of values of u, so that <r > 1. Hence a function, that
possesses an addition-theorem, cannot have only one doubly-infinite series of
values for its argument.

If cr be infinite, there is an infinite series of values of u of the form
+ p^ + p.flz ; an argument, similar to that in case (a), shews that this is,
in effect, the same as case (c).

It is obvious that cases (ii), (iii) and (iv) of § 152 are now completely
covered ; case (v) of § 152 is covered by case (c) now to be discussed in § 154.

154. For case (c), we have the series of values u represented by a number
of series of the form

where the quantities &> are not linearly equivalent to two quantities flj and
Q2- The original integer q is infinite.

Then, by §§ 108, 110, it follows that integers m can be chosen in an
unlimited variety of ways so that the modulus of

r=l

is infinitesimal, and therefore in the immediate vicinity of any point u^
there is an infinitude of points at which the function resumes its value.
Such a function would, as in previous instances, degenerate into a mere
constant ; and therefore the combination of values which gives rise to this
case does not occur.

All the possible cases have been considered: and the truth of Weierstrass's

20—2

308 EXAMPLES [154.

theorem* that a function, which has an algebraical addition-theorem, is either

imi

an algebraical function of u, or of e " (where &> is suitably chosen), or of g> (u),
where the periods of @(u) are suitably chosen, is established; and it has
incidentally been established — it is, indeed, essential to the derivation of the
theorem — that a function, which has an algebraical addition-theorem, has only
a finite number of values for a given argument.

It is easy to see that the first derivative has only a finite number of values
for a given argument; for the elimination of U between the algebraical
equations

, ,

leads to an equation in U' of the same finite degree as G in U.

Further, it is now easy to see that if the analytical function <£ (u), which
possesses an algebraical addition-theorem, be uniform, then it is a rational

iiru

function either of u, or of e w , or of $> (u) and $' (u) ; and that any uniform
function, which is transcendental in the sense of § 47 and which possesses an
algebraical addition-theorem, is either a simply-periodic function or a doubly-
periodic function.

The following examples will illustrate some of the inferences in regard to the number
of values of <p (u + v) arising from series of values for u and v.

Ex. I. Let U=u* + (2u+l)*.

Evidently m, the number of values of U for a value of u, is 4 ; and, as the rationalised
form of the equation is

the value of p, being the number of values of u for a given value of U, is 2. Thus the
equation in W should be, by § 151, of degree (4.22 — ) 16.

This equation is n {3 ( W2 - U2 - F2) + 1 - 2kr} = 0,

HI
where kr is any one of the eight values of

W(2W*-I)*+U(2U*-l$+V(2V*-l)*;

' •

an equation, when rationalised, of the 16th degree in W.

Ex. 2. Let U=cosu.

Evidently m = l; the values of u for a given value of U are contained in the double
series u + 2irn, -u + 2irn, for all values of n from -QO to +GO. The values of u + v are
, that is, u + v + 27rp; -u + 27rn+v + 2irm, that is, -u + v + 2-n-p ;
, that is, u-v + ^Trp; -u + 2irn-v + 2irm, that is, -u-v + Znp,

* The theorem has been used by Schwarz, Ges. Werke, t. ii, pp. 260—268, in determining all
the families of plane isothermic cirrves which are algebraical curves, an 'isothermic' curve being
of the form u = c, where w is a function satisfying the potential-equation

154.] THE DIFFERENTIAL EQUATION 309

to that the number of series of values of u+v is four, each series being simply-infinite.
It might thus be expected that the equation between U, V, W would be of degree

4 = ) 4 in W ; but it happens that

cos (u + v)=cos( -u-v),
and so the degree of the equation in W is reduced to half its degree. The equation is

W2 - 2 WU V+ U2 + V2 - 1 = 0.

Ex. 3. Let U=&iiu.

Evidently m = l; and there are two doubly-infinite series of values of u determined
by a given value of U, having the form u + 2ma> + 2m'<o', o> - w + 2mo> + 2m V. Hence the
values of u + v are

= u+v (mod. 2c0, 2o>') ; = ca-u + v (mod. 2«, 2«') ;
= ca + u-v(mod. 2o>, 2<o') ; = -u-v (mod. 2o>, 2&>') ;

four in number. The equation may therefore be expected to be of the fourth degree
in W; it is

4 (1 - 6T2) (1 - F2) (1 - IF2) = (2 - U2- F2- IF2 +£2*7272 W2^

155. But it must not be supposed that any algebraical equation between
U, V, W, which is symmetrical in U and V, is one necessarily implying the
representation of an algebraical addition-theorem. Without entering into a
detailed investigation of the formal characteristics of the equations that are
suitable, a latent test is given by implication in the following theorem, also
due to Weierstrass : —

If an analytical function possess an algebraical addition-theorem, an
algebraical equation involving the function and its first derivative with regard
to its argument exists ; and the coefficients in this equation do not involve the
argument of the function.

The proposition might easily be derived by assuming the preceding
proposition, and applying the known results relating to the algebraical
dependence between those functions, the types of which are suited to the
representation of the functions in question, and their derivatives ; we shall,
however, proceed more directly from the equation expressing the algebraical
addition-theorem in the form

G(U,V, F) = 0,

which may be regarded as a rationally irreducible equation.
Differentiating with regard to u, we have

WU'+MW^Q

dUL +dW '
and similarly, with regard to v, we have

a>+ *<=<>,

from which it follows that

310 EXPRESSION OF [155.

This equation* will, in general, involve W; in order to obtain an equation
free from W, we eliminate W between

n A a ^^ rr/ d6r Tr/

G = 0 and ^j- U' = „ V ,

oil ov

the elimination being possible because both equations are of finite degree;
and thus in any case we have an algebraical equation independent of W and
involving U, U', V, V.

Not more than one equation can arise by assigning various values to v, a
quantity that is independent of u ; for we should have either inconsistent
equations or simultaneous equations which, being consistent, determine a!
limited number of values of U and U' for all values of u, that is, only a
number of constants. Hence there can be only one equation, obtained by
assigning varying values to v; and this single equation is the algebraical
equation between the function and its first derivative, the coefficients being
independent of the argument of the function.

Note. A test of suitability of an algebraical equation G — 0 between
three variables U, V, W to represent an addition-theorem is given by the
condition that the elimination of W between

G-Q and U'^-V —

dU~ dV

leads to only a single equation between U and U' for different values of V
and V.

Ex. Consider the equation

(Z-U- V- W)*-4(1-U}(1- F)(l- F) = 0.
The deduced equation involving U1 and V is

(2FTF- V- W+ U} U' = (2UW- U- W+ V) V,

, th-it W (V-U}(V'+U'}

= (SV~lTUr

The elimination of W is simple. We have

_

(27-1) U'-(2U-\) F"

F U'-l-U V'

utd 2 U V W-«

(

Neglecting 4 (F+ U— 1) = 0, which is an irrelevant equation, arid multiplying by
(2F— 1) U' — (2U—l) F', which is not zero unless the numerator also vanish, and this
would make both U' and V zero, we have

( F+ U- 1) {(1 - F) U' - (1 - U} F'} 2 = (1 - U) (1 - F) ( U' - F') (2 F- 1) U' - (2 U- 1) F'},
and therefore V(U-V}(1- V] (7'2+ U( F- U} (1 - U} F'2 = 0.

It is permissible to adopt any subsidiary irrational or non-algebraical form as the equivalent
of G = 0, provided no special limitation to the subsidiary form be implicitly adopted. Thus, if W
can be expressed explicitly in terms of U and F, this resoluble (but irrational) equivalent of the
equation often leads rapidly to the equation between U and its derivative.

155.] THE ADDITION-THEOREM 311

When the irrelevant factor U- V is neglected, this equation gives

U'* F'2

U(l-U}~ V(l - V) '

the equation required : and this, indeed, is the necessary form in which the equation
involving U and U' arises in general, the variables being combined in associate pairs.
Each side is evidently a constant, say 4a2 ; and then we have

Then the value of U is sin2 (aM+/3), the arbitrary additive constant of integration
being /3 ; by substitution in the original equation, (3 is easily proved to be zero.

156. Again, if the elimination between

a - o and — U' - — V

aduu ~wv

be supposed to be performed by the ordinary algebraical process for finding

o/~y o/^r

the greatest common measure of G and U' %Tf — V %-\r> regarded as functions

of W, the final remainder is the eliminant which, equated to zero, is the
differential equation involving U, U', V, F'; and the greatest common measure,
equated to zero, gives the simplest equation in virtue of which the equations

G = 0 and ^y U' = _-^ V subsist. It will be of the form
oil ov

f(W,U,V, U',V') = 0.

If the function have only one value for each value of the argument, so that it
is a uniform function, this last equation can give only one value for W', for all
the other magnitudes that occur in the equation are uniform functions of
their respective arguments. Since it is linear in W, the equation can be
expressed in the form

W = R(U, V, U', V'\

where R denotes a rational function. Hence* : —

A uniform analytical function (f> (u), which possesses an algebraical
addition-theorem, is such that (f> (u + v) can be expressed rationally in terms
of $ (u), <£' (w), $ (v) and <j> (v).

It need hardly be pointed out that this result is not inconsistent with the
fact that the algebraical equation between (£ (u + v), (f> (u) and <f> (v) does not,
in general, express $(u + v) as a rational function of (f> (u) and <f>(v). And it
should be noticed that the rationality of the expression of <£ (u + v) in terms
of <j) (u), $ (v), (/>' (w), $ (v) is characteristic of functions with an algebraical
addition-theorem. Instances do occur of functions such that <j)(u + v) can be
expressed, not rationally, in terms of <£ (u), </> (v), </>' (u), </>' (v) ; they do not
possess an algebraical addition-theorem. Such an instance is furnished by
%(u)', the expression of £(u + v), given in Ex. 3 of § 131, can be modified so '
as to have the form indicated.

* The theorem is due to Weierstrass ; see Schwarz, § 2, (I.e. in note to p. 297).

CHAPTER XIV.

CONNECTION OF SURFACES.

157. IN proceeding to the discussion of multiform functions, it was
stated (§ 100) that there are two methods of special importance, one of which
is the development of Cauchy's general theory of functions of complex vari
ables and the other of which is due to Riemann. The former has been
explained in the immediately preceding chapters ; we now pass to the
consideration of Riemann's method. But, before actually entering upon it,
there are some preliminary propositions on the connection of surfaces which
must be established ; as they do not find a place in treatises on geometry, an
outline will be given here but only to that elementary extent which is
necessary for our present purpose.

In the integration of meromorphic functions, it proved to be convenient
to exclude the poles from the range of variation of the variable by means of
infinitesimal closed simple curves, each of which was thereby constituted a
limit of the region : the full boundary of the region was composed of the
aggregate of these non-intersecting curves.

Similarly, in dealing with some special cases of multiform functions, it
proved convenient to exclude the branch-points by means of infinitesimal
curves or by loops. And, in the case of the fundamental lemma of § 16, the
region over which integration extended was considered as one which possibly
had several distinct curves as its complete boundary.

These are special examples of a general class of regions, at all points
within the area of which the functions considered are monogeiiic, finite, and
continuous and, as the case may be, uniform or multiform. But, important
as are the classes of functions which have been considered, it is necessary to
consider wider classes of multiform functions and to obtain the regions which
are appropriate for the representation of the variation of the variable in each
case. The most conspicuous examples of such new functions are the algebraic
functions, adverted to in §§ 94 — 99 ; and it is chiefly in view of their value
and of the value of functions dependent upon them, as well as of the kind of
surface on which their variable can be simply represented, that we now
proceed to establish some of the topological properties of surfaces in general.

158. A surface is said to be connected when, from any point of it to any
other point of it, a continuous line can be drawn without passing out of the

158.]

EXAMPLES OF CONNECTED SURFACES

313

surface. Thus the surface of a circle, that of a plane ring such as arises in
Lambert's Theorem, that of a sphere, that of an anchor-ring, are connected
surfaces. Two non-intersecting spheres, not joined or bound together in any
manner, are not a connected surface but are two different connected surfaces.
It is often necessary to consider surfaces, which are constituted by an
aggregate of several sheets ; but, in order that the surface may be regarded
as connected, there must be junctions between the sheets.

One of the simplest connected surfaces is such a plane area as is enclosed
and completely bounded by the circumference of a circle. All lines drawn in
it from one internal point to another can be deformed into one another ; any
simple closed line lying entirely within it can be deformed so as to be
evanescent, without in either case passing over the circumference ; and any
simple line from one point of the circumference to another, when regarded as
an impassable barrier, divides the surface into two portions. Such a surface
is called* simply connected.

The kind of connected surface next in point of simplicity is such a plane
area as is enclosed between and is completely bounded by the circumferences
of two concentric circles. All lines in the surface
from one point to another cannot necessarily be
deformed into one another, e.g., the lines z0az and
zj)z; a simple closed line cannot necessarily be
deformed so as to be evanescent without crossing
the boundary, e.g., the line az^bza ; and a simple
line from a point in one part of the boundary to
a point in another and different part of the
boundary, such as a line AB, does not divide the
surface into two portions but, set as an impassable barrier, it makes the
surface simply connected.

Again, on the surface of an anchor-ring, a closed line can be drawn in
two essentially distinct ways, abc, cib'c', such
that neither can be deformed so as to be evanes
cent or so as to pass continuously into the other.
If abc be made the only impassable barrier, a
line such as afty cannot be deformed so as to be
evanescent ; if ab'c' be made the only impassable
barrier, the same holds of a line such as a/3'y'.
In order to make the surface simply connected,
two impassable barriers, such as abc and ab'c',
must be set.

Surfaces, like the flat ring or the anchor-

Fig. 35.

Fig. 36.

* Sometimes the term vionadelphic is used. The German equivalent is einfach ziisammen-
hangend.

314

CROSS-CUTS AND LOOP-CUTS

[158.

ring, are called* multiply connected] the establishment of barriers has made it
possible, in each case, to modify the surface into one which is simply connected.

159. It proves to be convenient to arrange surfaces in classes according
to the character of their connection ; and these few illustrations suggest that
the classification may be made to depend, either upon the resolution of the
surface, by the establishment of barriers, into one that is simply connected,
or upon the number of what may be called independent irreducible circuits.
The former mode — that of dependence upon the establishment of barriers —
will be adopted, thus following Biemann-f- ; but whichever of the two modes
be adopted (and they are not necessarily the only modes) subsequent de
mands require that the two be brought into relation with one another.

The most effective way of securing the impassability of a barrier is to
suppose the surface actually cut along the line of the barrier. Such a section
of a surface is either a cross-cut or a loop-cut.

If the section be made through the interior of the surface from one point

Fig. 37.

of the boundary to another point of the boundary, without intersecting itself
or meeting the boundary save at its extremities, it is called a cross-cut\.
Every part of it, as it is made, is to be regarded as boundary during the
formation of the remainder ; and any cross-cut, once made, is to be regarded
as boundary during the formation of any cross-cut subsequently made.
Illustrations are given in Fig. 37.

The definition and explanation imply that the surface has a boundary.
Some surfaces, such as a complete sphere and a complete anchor-ring, do not
possess a boundary; but, as will be seen later (§§ 163, 168) from the
discussion of the evanescence of circuits, it is desirable to assign some
boundary in order to avoid merely artificial difficulties as to the numerical

* Sometimes the term polyadc.lphic is used. The German equivalent is mehrfach zusammen-
Mngcnd.

t " Grundlagen fur eine allgemeine Theorie der Functionen einer veriindeiiichen complexen
Grosse," Eiemann's Gesammelte Werke, pp. 9 — 12; "Theorie der Abel'schen Functionen," ib.,/
pp. 84—89. When reference to either of these memoirs is made, it will be by a citation "et ih^
page or pages in the volume of lliemann's Collected Works.

£ This is the equivalent used for the German word Querschnitt ; French writers use Section,
and Italian writers use Trasversale or Taglio trasversale.

159.] CONNECTION DEFINED 315

expression of the connection. This assignment usually is made by taking for
the boundary of a surface, which otherwise has no boundary, an infinitesimal
closed curve, practically a point; thus in the figure of the anchor-ring
(Fig. 36) the point a is taken as a boundary, and each of the two cross-cuts
begins and ends in a.

If the section be made through the interior of the surface from a point
not on the boundary and, without meeting the boundary or crossing itself,
return to the initial point, (so that it has the form of a simple curve lying

Fig. 38.

entirely in the surface), it is called* a loop-cut. Thus a piece can be cut
out of a bounded spherical surface by a loop-cut (Fig. 38) ; but it does
not necessarily give a separate piece when made in the surface of an
anchor-ring.

It is evident that both a cross-cut and a loop-cut furnish a double
boundary-edge to the whole aggregate of surface, whether consisting of two
pieces or of only one piece after the section.

Moreover, these sections represent the impassable barriers of the pre
liminary explanations ; and no specified form was assigned to those barriers.
It is thus possible, within certain limits, to deform a cross-cut or a loop-cut
continuously into a closely contiguous and equivalent position. If, for
instance, two barriers initially coincide over any finite length, one or other
can be slightly deformed so that finally they intersect only in a point ; the
same modification can therefore be made in the sections.

The definitions of simple connection and of multiple connection will nowf*
be as follows : —

A surface is simply connected, if it be resolved into two distinct pieces by
every cross-cut; but if there be any cross-cut, which does not resolve it into
distinct pieces, the surface is multiply connected.

160. Some fundamental propositions, relating to the connection of
surfaces, may now be derived.

* This is the equivalent used for the German word Riickkehrsclmitt ; French writers use the
word Retroscction.

t Other definitions will be required, if the classification of surfaces be made to depend on
methods other than resolution by sections.

316 RESOLUTION BY CROSS-CUTS [160.

I. Each of the two distinct pieces, into which a simply connected surface S
is resolved by a cross-cut, is itself simply connected.

If either of the pieces, made by a cross-cut ab, be not simply connected,
then some cross-cut cd must be possible which will not resolve that piece into
distinct portions.

If neither c nor d lie on ab, then the obliteration of the cut ab will restore
the original surface 8, which now is not resolved by the cut cd into distinct
pieces.

If one of the extremities of cd, say c, lie on ab, then the obliteration of the
portion cb will change the two pieces into a single piece which is the original
surface 8; and 8 now has a cross-cut acd, which does not resolve it into
distinct pieces.

If both the extremities lie on ab, then the obliteration of that part of ab
which lies between c and d will change the two pieces into one ; this is the
original surface 8, now with a cross-cut acdb, which does not resolve it into
distinct pieces.

These are all the possible cases should either of the distinct pieces of 8
not be simply connected ; each of them leads to a contradiction of the simple
connection of 8', therefore the hypothesis on which each is based is untenable,
that is, the distinct pieces of 8 in all the cases are simply connected.

COROLLARY 1. A singly connected surface is resolved by n cross-cuts into
Ti+1 distinct pieces, each simply connected; and an aggregate of m simply
connected surfaces is resolved by n cross-cuts into n -f m distinct pieces each
simply connected.

COROLLARY 2. A surface that is resolved into two distinct simply con
nected pieces by a cross-cut is simply connected before the resolution.

COROLLARY 3. // a multiply connected surface be resolved into two
different pieces by a cross-cut, both of these pieces cannot be simply connected.

We now come to a theorem* of great importance : —

II. If a resolution of a surface by m cross-cuts into n distinct simply
connected pieces be possible, and also a different resolution of the same surface by
fjb cross-cuts into v distinct simply connected pieces, then m — n = fj, — v.

Let the aggregate of the n pieces be denoted by 8 and the aggregate of
the v pieces by 2 : and consider the effect on the original surface of a united
system of in + p simultaneous cross-cuts made up of the two systems of the
m and of the /j, cross-cuts respectively. The operation of this system can be
carried out in two ways : (i) by effecting the system of /u, cross-cuts on 8 and

* The following proof of this proposition is substantially due to Neumann, p. 157. Another
proof is given by Riemann, pp. 10, 11, and is amplified by Durege, Elemente der Theorie der
Functional, pp. 183 — 190 ; and another by Lippich, see Durege, pp. 190 — 197.

160.] CONNECTIVITY 317

(ii) by effecting the system of m cross-cuts on 2 : with the same result on the
original surface.

After the explanation of § 159, we may justifiably assume that the lines
of the two systems of cross-cuts meet only in points, if at all : let 8 be the
number of points of intersection of these lines. Whenever the direction of a
cross-cut meets a boundary line, the cross-cut terminates ; and if the direction
continue beyond that boundary line, that produced part must be regarded as
a new cross-cut.

Hence the new system of /u, cross-cuts applied to S is effectively equiva
lent to (j, + & new cross-cuts. Before these cuts were made, S was composed
of n simply connected pieces ; hence, after they are applied, the new arrange
ment of the original surface is made up of n + (/j, + 8) simply connected
pieces.

Similarly, the new system of m cross-cuts applied to 2 will give an
arrangement of the original surface made up of v + (m + 8) simply connected
pieces. These two arrangements are the same : and therefore

n + fj, + 8 — v + in + 8,
so that m — n = p — v.

It thus appears that, if by any system of q cross-cuts a multiply connected
surface be resolved into a number p of pieces distinct from one another and
all simply connected, the integer q — p is independent of the particular
system of the cross-cuts and of their configuration. The integer q—p is
therefore essentially associated with the character of the multiple connection
of the surface : and its invariance for a given surface enables us to arrange
surfaces according to the value of the integer.

No classification among the multiply connected surfaces has yet been
made : they have merely been defined as surfaces in which cross-cuts can be
made that do not resolve the surface into distinct pieces.

It is natural to arrange them in classes according to the number of cross
cuts which are necessary to resolve the surface into one of simple connection
or a number of pieces each of simple connection.

For a simply connected surface, no such cross-cut is necessary: then
q = 0, p=l, and in general q — p = — l. We shall say that the connectivity*
is unity. Examples are furnished by the area of a plane circle, and by a
spherical surface with one hole^.

A surface is called doubly- connected when, by one appropriate cross-cut,
the surface is changed into a single surface of simple connection : then q = 1,
p = 1 for this particular resolution, and therefore in general, q—p = Q. We

* Sometimes order of connection, sometimes adelphic order ; the German word, that is used,
is Grundzahl.

+ The hole is made to give the surface a boundary (§ 163).

318 EFFECT OF CROSS-CUTS [160.

shall say that the connectivity is 2. Examples are furnished by a plane ring
and by a spherical surface with two holes.

A surface is called triply-connected when, by two appropriate cross-cuts,
the surface is changed into a single surface of simple connection : then q = 2,
p = l for this particular resolution and therefore, in general, q — p = l. We
shall say that the connectivity is 3. Examples are furnished by the surface
of an anchor- ring with one hole in it*, and by the surfaces -f- in Figure 39, the
surface in (2) not being in one plane but one part beneath another.

Fig. 39.

And, in general, a surface will be said to be ^V-ply connected or its
connectivity will be denoted by N, if, by N — 1 appropriate cross-cuts, it can
be changed into a single surface that is simply connected |. For this
particular resolution q = N—\, p = l: and therefore in general

q-p = N-2,
or N = q-p + 2.

Let a cross-cut I be drawn in a surface of connectivity N. There are
two cases to be considered, according as it does not or does divide the surface
into distinct pieces.

First, let the surface be only one piece after I is drawn : and let its
connectivity then be N'. If in the original surface q cross-cuts (one of
which can, after the preceding proposition, be taken to be I) be drawn
dividing the surface into p simply connected pieces, then

N = q-p+ 2.

To obtain these p simply connected pieces from the surface after the cross-cut
I, it is evidently sufficient to make the q — 1 original cross-cuts other than I ;
that is, the modified surface is such that by q — 1 cross-cuts it is resolved into
p simply connected pieces, and therefore

Hence N' = N — 1, or the connectivity of the surface is diminished by unity.

* The hole is made to give the surface a boundary (§ 163).

t Riemann, p. 89.

J A few writers estimate the connectivity of such a surface as N- 1, the same as the number
of cross-cuts which can change it into a single surface of the simplest rank of connectivity : the
estimate in the text seems preferable.

160.]

ON THE CONNECTIVITY

319

Secondly, let the surface be two pieces after I is drawn, of connectivities
Ni and N2 respectively. Let the appropriate JVj — 1 cross-cuts in the former,
and the appropriate N2 — 1 in the latter, be drawn so as to make each a
simply connected piece. Then, together, there are two simply connected
pieces.

To obtain these two pieces from the original surface, it will suffice to
make in it the cross-cut I, the Ni — I cross-cuts, and the N2—l cross-cuts,
that is, 1 + (Ni. — 1) + (N* — 1) or Nj, + N2 — 1 cross-cuts in all. Since these,
when made in the surface of connectivity N, give two pieces, we have

and therefore

If one of the pieces be simply connected, the connectivity of the other is JV;
so that, if a simply connected piece of surface be cut off a multiply connected
surface, the connectivity of the remainder is unchanged. Hence :

III. If a cross-cut be made in a surface of connectivity N and if it do
not divide it into separate pieces, the connectivity of the modified surface is
N—l; but if it divide the surface into two separate pieces of connectivities N!
and N«, then Nl + N2 = N+ 1.

Illustrations are shewn, in Fig. 40, of the effect of cross-cuts on the two
surfaces in Fig. 39.

IV. In the same way it may be proved that, if s cross-cuts be made in a
surface of connectivity N and divide it into r+l separate pieces (where r^.s)
of connectivities N1} N2, ..., Nr+l respectively, then

a more general result including both of the foregoing cases.

Thus far we have been considering only cross-cuts : it is now necessary
to consider loop-cuts, so far as they affect the connectivity of a surface in
which they are made.

320 EFFECT OF LOOP-CUTS [160.

A loop-cut is changed into a cross-cut, if from A any point of it a cross-cut
be made to any point C in a boundary-curve of the
original surface, for CAbdA (Fig. 41) is then evi- /•

dently a cross-cut of the original surface ; and CA is
a cross-cut of the surface, which is the modification
of the original surface after the loop-cut has been
made. Since, by definition, a loop-cut does not
meet the boundary, the cross-cut CA does not
divide the modified surface into distinct pieces ;
hence, according as the effect of the loop-cut is, \ Fi8- 41-

or is not, that of making distinct pieces, so will
the effect of the whole cross-cut be, or not be, that of making distinct pieces.

161. Let a loop-cut be drawn in a surface of connectivity N; as before
for a cross-cut, there are two cases for consideration, according as the loop-cut
does or does not divide the surface into distinct pieces.

First, let it divide the surface into two distinct pieces, say of connectivities
N! and N2 respectively. Change the loop-cut into a cross-cut of the original
surface by drawing a cross-cut in either of the pieces, say the second, from a
point in the course of the loop-cut to some point of the original boundary.
This cross-cut, as a section of that piece, does not divide it into distinct
pieces: and therefore the connectivity is now N? (= N2 — 1). The effect of
the whole section, which is a single cross-cut, of the original surface is to
divide it into two pieces, the connectivities of which are JVa and N2' : hence,
by S 160, III.,

and therefore N1 + Na

If the piece cut out be simply connected, say JVj. = 1, then the connectivity
of the remainder is N + 1. But such a removal of a simply connected piece
by a loop-cut is the same as making a hole in a continuous part of the
surface : and therefore the effect of making a simple hole in a continuous part
of a surface is to increase by unity the connectivity of the surface.

If the piece cut out be doubly connected, say N: = 2, then the connect
ivity of the remainder is N, the same as the connectivity of the original
surface. Such a portion would be obtained by cutting out a piece with a
hole in it which, so far as concerns the original surface, would be the same as
merely enlarging the hole — an operation that naturally would not affect
the connectivity.

Secondly, let the loop -cut not divide the surface into two distinct pieces :
and let N' be the connectivity of the modified surface. In this modified
surface make a cross-cut k from any point of the loop-cut to a point of the
boundary: this does not divide it into distinct pieces and therefore the
connectivity after this last modification is N' -I. But the surface thus

161.] ON THE CONNECTIVITY 321

finally modified is derived from the original surface by the single cross-cut,
constituted by the combination of k with the loop-cut : this single cross-cut
does not divide the surface into distinct pieces and therefore the connectivity
after the modification is N — 1. Hence

that is, JV' = N, or the connectivity of a surface is not affected by a loop-cut
which does not divide the surface into distinct pieces.

Both of these results are included in the following theorem : —

V. If after any number of loop-cuts made in a surface of connectivity
N, there be r + 1 distinct pieces of surface, of connectivities JV^ JV2, ..., Nr+lt
then

N, + N3 + ...... + JVr+1 = JV+2r.

Let the number of loop-cuts be s. Each of them can be changed into a
cross-cut of the original surface, by drawing in some one of the pieces, as may
be convenient, a cross-cut from a point of the loop-cut to a point of a
boundary ; this new cross-cut does not divide the piece in which it is drawn
into distinct pieces. If k such cross-cuts (where k may be zero) be drawn in
the piece of connectivity Nm, the connectivity becomes Nm', where

N ' — N~ — If-

•" m — •*•• m I" j
r+l r+l r+l

hence 2 Nm' = 2 Nm-2k= X Nm - s.

m=\ m-\ m=l

We now have s cross-cuts dividing the surface of connectivity JV into r + l
distinct pieces, of connectivities JV/, JV/, ..., JV/, Nr+1' ; and therefore, by
§ 160, IV.,

so that JVj + JV2 + . . . 4- Nr+1 = JV + 2r.

This result could have been obtained also by combination and repetition
of the two results obtained for a single loop-cut.

Thus a spherical surface with one hole in it is simply connected : when
n — l other different holes* are made in it, the edges of the holes being
outside one another, the connectivity of the surface is increased by n— 1,
that is, it becomes n. Hence a spherical surface with n holes in it is n-ply
connected.

162. Occasionally, it is necessary to consider the effect of a slit made in
the surface.

If the slit have neither of its extremities on a boundary (and therefore no
point on a boundary) it can be regarded as the limiting form of a loop-cut
which makes a hole in the surface. Such a slit therefore (§ 161) increases the
connectivity by unity.

* These are holes in the surface, not holes bored through the volume of the sphere ; one of
the latter would give two holes in the surface.

F- 21

BOUNDARIES [162.

If the slit have one extremity (but no other point) on a boundary, it can
be regarded as the limiting form of a cross-cut, which returns
on itself as in the figure, and cuts off a single simply con- /
nected piece. Such a slit therefore (§ 160, III.) leaves the
connectivity unaltered.

If the slit have both extremities on boundaries, it ceases \
to be merely a slit : it is a cross-cut the effect of which on Fl8- 42-

the connectivity has been obtained. We do not regard such
sections as slits.

163. In the preceding investigations relative to cross-cuts and loop-cuts,
reference has continually been made to the boundary of the surface con
sidered.

The boundary of a surface consists of a line returning to itself, or of a
system of lines each returning to itself. Each part of such a boundary-line
as it is drawn is considered a part of the boundary, and thus a boundary-line
cannot cut itself and pass beyond its earlier position, for a boundary cannot
be crossed: each boundary-line must therefore be a simple curve*.

Most surfaces have boundaries : an exception arises in the case of closed
surfaces whatever be their connectivity. It was stated (§ 159) that a
boundary is assigned to such a surface by drawing an infinitesimal simple
curve in it or, what is the same thing, by making a small hole. The
advantage of this can be seen from the simple example of a spherical
surface.

When a small hole is made in any surface the connectivity is increased
by unity : the connectivity of the spherical surface after the hole is made is
unity, and therefore the connectivity of the complete spherical surface
must be taken to be zero.

The mere fact that the connectivity is less than unity, being that of the
simplest connected surfaces with which we have to deal,
is not in itself of importance. But let us return for a
moment to the suggested method of determining the
connectivity by means of the evanescence of circuits
without crossing the boundary. When the surface is
the complete spherical surface (Fig. 43), there are two
essentially distinct ways of making a circuit C evan
escent, first, by making it collapse into the point a, Fig. 43.
secondly by making it expand over the equator and

then collapse into the point b. One of the two is superfluous : it introduces
an element of doubt as to the mode of evanescence unless that mode be
specified a specification which in itself is tantamount to an assignment of

* Also a line not returning to itself may be a boundary ; it can be regarded as the limit of a
simple curve when the area becomes infinitesimal.

163.] EFFECT OF CROSS-CUTS ON BOUNDARIES 323

boundary. And in the case of multiply connected surfaces the absence of
boundary, as above, leads to an artificial reduction of the connectivity by
unity, arising not from the greater simplicity of the surface but from the
possibility of carrying out in two ways the operation of reducing any circuit
to given circuits, which is most effective when only one way is permissible.
We shall therefore assume a boundary assigned to such closed surfaces as in
the first instance are destitute of boundary.

164. The relations between the number of boundaries and the connect
ivity of a surface are given by the following propositions.

I. The boimdary of a simply connected surface consists of a single line.

When a boundary consists of separate lines, then a cross-cut can be made
from a point of one to a point of another. By proceeding from
P, a point on one side of the cross-cut, along the boundary
ac...cVwe can by a line lying wholly in the surface reach a
point Q on the other side of the cross-cut : hence the parts of
the surface on opposite sides of the cross-cut are connected.
The surface is therefore not resolved into distinct pieces by the
cross-cut.

A simply connected surface is resolved into distinct pieces Fig. 44.
by each cross-cut made in it : such a cross-cut as the foregoing
is therefore not possible, that is, there are not separate lines which make up
its boundary. It has a boundary : the boundary therefore consists of a single
line.

II. A cross-cut either increases by unity or diminishes by unity the number
of distinct boundary -lines of a multiply connected surface.

A cross-cut is made in one of three ways : either from a point a of one
boundary-line A to a, point b of another boundary-line B ; or from a point a
of a boundary-line to another point a' of the same boundary-line ; or from a
point of a boundary-line to a point in the cut itself.

If made in the first way, a combination of one edge of the cut, the
remainder of the original boundary A, the other edge of the cut and the
remainder of the original boundary B taken in succession, form a single
piece of boundary ; this replaces the two boundary-lines A and B which
existed distinct from one another before the cross-cut was made. Hence the
number of lines is diminished by unity. An example is furnished by a plane
ring (ii., Fig. 37, p. 314).

If made in the second way, the combination of one edge of the cut with
the piece of the boundary on one side of it makes one boundary-line, and the
combination of the other edge of the cut with the other piece of the boundary
makes another boundary-line. Two boundary-lines, after the cut is made,

21—2

324 NUMBER OF BOUNDARY-LINES [164.

replace a single boundary-line, which existed before it was made : hence the
number of lines is increased by unity. Examples are furnished by the cut
surfaces in Fig. 40, p. 319.

If made in the third way, the cross-cut may be considered as constituted
by a loop-cut and a cut joining the loop-cut to the boundary. The boundary-
lines may now be considered as constituted (Fig. 41, p. 320) by the closed
curve ABD and the closed boundary abda'c'e'...eca; that is, there are now
two boundary-lines instead of the single boundary-line ce...e'c'c in the uncut
surface. Hence the number of distinct boundary-lines is increased by unity.

COROLLARY. A loop-cut increases the number of distinct boundary-lines
by two.

This result follows at once from the last discussion.

III. The number of distinct boundary-lines of a surface of connectivity N
is N — 2k, where k is a positive integer that may be zero.

Let m be the number of distinct boundary-lines ; and let N — 1 appro
priate cross-cuts be drawn, changing the surface into a simply connected
surface. Each of these cross-cuts increases by unity or diminishes by unity
the number of boundary-lines ; let these units of increase or of decrease be
denoted by e^ e2, ..., €#_!. Each of the quantities e is + 1 ; let k of them be
positive, and N — 1 — k negative. The total number of boundary-lines is

therefore

m + k-(N-l-k).

The surface now is a single simply connected surface, and there is therefore
only one boundary-line ; hence

m + k-(N-l-k) = l,
so that m = N — 2k ;

and evidently k is an integer that may be zero.

COROLLARY 1. A closed surface with a single boundary-line* is of odd
connectivity.

For example, the surface of an anchor-ring, when bounded, is of con
nectivity 3; the surface, obtained by boring two holes through the volume
of a solid sphere, is, when bounded, of connectivity 5.

If the connectivity of a closed surface with a single boundary be 2p + 1,
the surface is often said-f- to be of class p (§ 178, p. 349.)

COROLLARY 2. If the number of distinct boundary lines of a surface of
connectivity N be N, any loop-cut divides the surface into two distinct pieces.

After the loop-cut is made, the number of distinct boundary-lines is
N+2; the connectivity of the whole of the cut surface is therefore not less

* See § 159.

t The German word is Geschlecht ; French writers use the word genre, and Italians genere.

164.] LHUILIER'S THEOREM 325

than N+2. It has been proved that a loop-cut, which does not divide the
surface into distinct pieces, does not affect the connectivity ; hence as the
connectivity has been increased, the loop-cut must divide the surface into
two distinct pieces. It is easy, by the result of § 161, to see that, after the
loop-cut is made, the sum of connectivities of the two pieces is N+2, so
that the connectivity of the whole of the cut surface is equal to N + 2.

Note. Throughout these propositions, a tacit assumption has been made,
which is important for this particular proposition when the surface is the
means of representing the variable. The assumption is that the surface is
bifacial and not unifacial ; it has existed implicitly throughout all the
geometrical representations of variability : it found explicit expression in
§ 4 when the plane was brought into relation with the sphere : and a cut
in a surface has been counted a single cut, occurring in one face, though it
would have to be counted as two cuts, one on each side, were the surface
unifacial.

The propositions are not necessarily valid, when applied to unifacial
surfaces. Consider a surface made out of a long rectangular slip of paper,
which is twisted once (or any odd number of times) and then has its ends
fastened together. This surface is of double connectivity, because one
section can be made across it which does not divide it into separate pieces ;
it has only a single boundary-line, so that Prop. III. just proved does not
! apply. The surface is unifacial ; and it is possible, without meeting the
boundary, to pass continuously in the surface from a point P to another
point Q which could be reached merely by passing through the material
at P.

We therefore do not retain unifacial surfaces for consideration.

165. The following proposition, substantially due to Lhuilier*, may be
taken in illustration of the general theory.

If a closed surface of connectivity 2N + 1 (or of class N) be divided by
circuits into any number of simply connected portions, each in the form of a
curvilinear polygon, and if F be the number of polygons, E be the number of
edges and S the number of angular points, then

2N=2 + JE-F-S.

Let the edges E be arranged in systems, a system being such that any
lino in it can be reached by passage along some other line or lines of the
system ; let k be the number of such systems -f. To resolve the surface into a
number of simply connected pieces composed of the F polygons, the cross-cuts
will be made along the edges ; and therefore, unless a boundary be assigned

* Gergonne, Ann. de Math., t. iii, (1813), pp. 181—186; see also Mobius, Ges. Werke, t. ii,
p. 468. A circuit is defined in § 166.

t The value of k is 1 for the proposition and is greater than 1 for the Corollary.

326 LHUILIER'S THEOREM [165.

to the surface in each system of lines, the first cut for any system will be a
loop-cut. We therefore take k points, one in each system as a boundary ;
the first will be taken as the natural boundary of the surface, and the
remaining k—\, being the limiting forms of k — 1 infinitesimal loop-cuts,
increase the connectivity of the surface by k — 1, that is, the connectivity now
is 2N+k.

The result of the cross-cuts is to leave F simply connected pieces : hence
Q, the number of cross-cuts, is given by

At every angular point on the uncut surface, three or more polygons are
contiguous. Let Sm be the number of angular points, where m polygons are
contiguous; then

Again, the number. of edges meeting at each of the S3 points is three, atl
each of the $4 points is four, at each of the $5 points is five, and so on ; hence,
in taking the sum 3$3 + 4$4 + 5$5 + . . ., each edge has been counted twice, once
for each extremity. Therefore

Consider the composition of the extremities of the cross-cuts ; the number
of the extremities is 2Q, twice the number of cross-cuts.

Each of the k points furnishes two extremities; for each such point
is a boundary on which the initial cross-cut for each of the systems must
begin and must end. These points therefore furnish 2k extremities.

The remaining extremities occur in connection with the angular points.
In making a cut, the direction passes from a boundary along an edge, past
the point along another edge and so on, until a boundary is reached ; so that
on the first occasion when a cross-cut passes through a point, it is made along
two of the edges meeting at the point. Every other cross-cut passing through
that point must begin or end there, so that each of the S3 points will furnish
one extremity (corresponding to the remaining one cross-cut through the
point), each of the $4 points will furnish two extremities (corresponding to
the remaining two cross-cuts through the point), and so on. The total
number of extremities thus provided is

S3 + 2St+3S5 + ...
Hence 2Q = 2k + 83 + 2St + 3S6+ ...

or Q = k + E-S,

which combined with Q = 2N + k + F - 2,
leads to the relation 2N=2 + E-F-S.

165.] CIRCUITS ON CONNECTED SURFACES 327

The simplest case is that of a sphere, when Euler's relation F + S =• E + 2
is obtained. The case next in simplicity is that of an anchor-ring, for which
the relation is F+ S = E.

COROLLARY. If the result of making the cross-cuts along the various edges
be to give the F polygons, not simply connected areas but areas of connectivities
jYj + 1, jV2 + l, ..., Np+1 respectively, then the connectivity of the original
surface is given by

166. The method of determining the connectivity of a surface by means
of a system of cross-cuts, which resolve it into one or more simply connected
pieces, will now be brought into relation with the other method, suggested
in § 159, of determining the connectivity by means of irreducible circuits.

A closed line drawn on the surface is called a circuit.

A circuit, which can be reduced to a point by' continuous deformation
without crossing the boundary, is called reducible ; a circuit, which cannot be
so reduced, is called irreducible.

An irreducible circuit is either (i) simple, when it cannot without crossing
the boundary be deformed continuously into repetitions of one or more
circuits ; or (ii) multiple, when it can without crossing the boundary be
deformed continuously into repetitions of a single circuit ; or (iii) compound,
when it can without crossing the boundary be deformed continuously into
combinations of different circuits, that may be simple or multiple. The
distinction between simple circuits and compound circuits, that involve no
multiple circuits in their combination, depends upon conventions adopted for
each particular case.

A circuit is said to be reconcileable with the system of circuits into a
combination of which it can be continuously deformed.

If a system of circuits be reconcileable with a reducible circuit, the
system is said to be reducible.

As there are two directions, one positive and the other negative, in which
a circuit can be described, and as there are possibilities of repetitions and of
compositions of circuits, it is clear that circuits can be represented by linear
algebraical expressions involving real quantities and having merely numerical
coefficients.

Thus a reducible circuit can be denoted by 0.

If a simple irreducible circuit, positively described, be denoted by a, the
same circuit, negatively described, can be denoted by — a.

The multiple circuit, which is composed of m positive repetitions of the
simple irreducible circuit a, would be denoted by ma ; but if the m repetitions
were negative, the multiple circuit would be denoted by — ma.

328 CIRCUITS [106.

A compound circuit, reconcileable with a system of simple irreducible
circuits a1} a2, ..., an would be denoted by m1a1 + m2a2-\- ... + mnan, where
mj, m2, ..., mn are positive or negative integers, being the net number of
positive or negative descriptions of the respective simple irreducible circuits.

The condition of the reducibility of a system of circuits al, «2, ..., an,
each one of which is simple and irreducible, is that integers m1} m.2, ..., mn
should exist such that

m^j + m2a2 + . . . + mnan = 0,

the sign of equality in this equation, as in other equations, implying that
continuous deformation without crossing the boundary can change into one
another the circuits, denoted by the symbols on either side of the sign.

The representation of any compound circuit in terms of a system of
independent irreducible circuits is unique : if there were two different
expressions, they could be equated in the foregoing sense and this would
imply the existence of a 'relation

P& + p.2a2 + . . . +pnan = 0,
which is excluded by the fact that the system is irreducible.

Further, equations can be combined linearly, provided that the coefficients
of the combinations be merely numerical.

167. In order, then, to be in a position to estimate circuits on a multiply
connected surface, it is necessary that an irreducible system of irreducible
simple circuits should be known, such a system being considered complete
when every other circuit on the surface is reconcileable with the system.

Such a system is not necessarily unique ; and it must be proved that, if
more than one complete system be obtainable, any circuit can be reconciled with
each system.

First, the number of simple irreducible circuits in any complete system
must be tlie same for the same surface.

Let a1} ..., ap; and b1} ..., bn; be two complete systems. Because a1} ...,
ap constitute a complete system, every circuit of the system of circuits b is
reconcileable with it ; that is, integers ra# exist, such that

br = mlral + m.2ra.2 + . . . + mprap,

for r = 1, 2, ..., n. If n were >p, then by combining linearly each equation
after the first p equations with those p equations, and eliminating al, ..., ap
from the set of p + 1 equations, we could derive n —p relations of the form

M^ + M,b2 + . . . + Mnbn = 0,

where the coefficients M, being determinants the constituents of which are
integers, would be integers. The system of circuits b is irreducible, and there
are therefore no such relations ; hence n is not greater than p.

167.] ON CONNECTED SURFACES 329

Similarly, by considering the reconciliation of each circuit a with the
irreducible system of circuits b, it follows that p is not greater than n.

Hence p and n are equal to one another. And, because each system is a
complete system, there are integers A and B such that
ar = Arlbi + Ar2b.2 4- • • • + Arnbn (r = I, ...,
bs = Bg^ + Bs.2a2 + . . . -I- BmOn (s = l, ...,

The determinant of the integers A is equal to + 1 ; likewise the deter
minant of the integers B.

Secondly, let x be a circuit reconcileable with the system of circuits a : it is
reconcileable with any other complete system of circuits.

Since x is reconcileable with the system a, integers m1} ..., mn can be

found such that

x = ??i1«1 + . . . + mnan.

Any other complete system of n circuits b is such that the circuits a can
be expressed in the form

ar = Anbj. + ... + Arnbn , (r = 1, . . ., n),
where the coefficients A are integers ; and therefore

n n n

x = b1'2 mrArl 4- 62 S mrArz + . . . + bn X mrArn

r=l r=l r=l

= gri&i + gr2&a + ~'+qnl>n,

where the coefficients q are integers, that is, x is reconcileable with the
complete system of circuits b.

168. It thus appears that for the construction of any circuit on a surface,
it is sufficient to know some one complete system of simple irreducible
circuits. A complete system is supposed to contain the smallest possible
number of simple circuits : any one which is reconcileable with the rest is
omitted, so that the circuits of a system may be considered as independent.
Such a system is indicated by the following theorems : —

I. No irreducible simple circuit can be drawn on a simply connected
surface*.

If possible, let an irreducible circuit G be drawn in a simply connected
surface with a boundary B. Make a loop-cut along C, and change it into a
cross-cut by making a cross-cut A from some point of C to a point of B ;
this cross-cut divides the surface into two simply connected pieces, one of
which is bounded by B, the two edges of A, and one edge of the cut along C,
and the other of which is bounded entirely by the cut along C.

The latter surface is smaller than the original surface ; it is simply

connected and has a single boundary. If an irreducible simple circuit can

be drawn on it, we proceed as before, and again obtain a still smaller simply

connected surface. In this way, we ultimately obtain an infinitesimal

* All surfaces considered are supposed to be bounded.

330 RELATIONS BETWEEN CONNECTIVITY [168.

element ; for every cut divides the surface, in which it is made, into
distinct pieces. Irreducible circuits cannot be drawn in this element ; and
therefore its boundary is reducible. This boundary is a circuit in a larger
portion of the surface : the circuit is reducible so that, in that larger portion
no irreducible circuit is possible and therefore its boundary is reducible.
This boundary is a circuit in a still larger portion, and the circuit is
reducible : so that in this still larger portion no irreducible circuit is possible
and once more the boundary is reducible.

Proceeding in this way, we find that no irreducible simple circuit is
possible in the original surface.

COROLLARY. No irreducible circuit can be drawn on a simply connected
surface.

II. A complete system of irreducible simple circuits for a surface of
connectivity N contains N— I simple circuits, so that every other circuit on the
surface is reconcileable with that system.

Let the surface be resolved by cross-cuts into a single simply connected
surface: N— 1 cross-cuts will be necessary. Let CD be
any one of them : and let a and b be two points on the /e

opposite edges of the cross-cut. Then since the surface is L n

simply connected, a line can be drawn in the surface from
a to b without passing out of the surface or without
meeting a part of the boundary, that is, without meeting
any other cross-cut. The cross-cut CD ends either in Fis- 45-

another cross-cut or in a boundary; the line ae...fb
surrounds that other cross-cut or that boundary as the case may be : hence,
if the cut CD be obliterated, the line ae...fba is irreducible on the surface in
which the other N — 2 cross-cuts are made. But it meets none of those cross
cuts; hence, when they are all obliterated so as to restore the unresolved
surface of connectivity N, it is an irreducible circuit. It is evidently riot
a repeated circuit; hence it is an irreducible simple circuit. Hence the
line of an irreducible simple circuit on an unresolved surface is given by
a line passing from a point on one edge of a cross-cut in the resolved
surface to a point on the opposite edge.

Since there are N -I cross-cuts, it follows that N —1 irreducible simple
circuits can thus be obtained: one being derived in the foregoing manner
from each of the cross-cuts, which are necessary to render the surface simply
connected. It is easy to see that each of the irreducible circuits on an
unresolved surface is, by the cross-cuts, rendered impossible as a circuit on
the resolved surface.

But every other irreducible circuit C is reconcileable with the N—l
circuits, thus obtained. If there be one not reconcileable with these N-l
circuits, then, when all the cross-cuts are made, the circuit C is not rendered

168.]

AND IRREDUCIBLE CIRCUITS

331

impossible, if it be not reconcileable with those which are rendered impossible
by the cross-cuts : that is, there is on the resolved surface an irreducible
circuit. But the resolved surface is simply connected, and therefore no
irreducible circuit can be drawn on it : hence the hypothesis as to C, which
leads to this result, is not tenable.

Thus every other circuit is reconcileable with the system of N — 1 circuits :
and therefore the system is complete*.

This method of derivation of the circuits at once indicates how far a
system is arbitrary. Each system of cross-cuts leads to a complete system of
irreducible simple circuits, and vice versa ; as the one system is not unique,
so the other system is not unique.

For the general question, Jordan's memoir, Des contours traces sur les surfaces,
Liouville, 2me Ser., t. xi., (1866), pp. 110—130, may be consulted.

Ex. 1. On a doubly connected surface, one irreducible simple circuit can be drawn.
It is easily obtained by first resolving the surface into one that is simply connected —
a single cross-cut CD is effective for this purpose — and then by drawing a curve aeb in the

Fig. 46, (i).

surface from one edge of the cross-cut to the other. All other irreducible circuits on the
unresolved surface are reconcileable with the circuit aeba.

Ex. 2. On a triply- connected surface, two independent irreducible circuits can be

Fig. 46, (ii).

* If the number of independent irreducible simple circuits be adopted as a basis for the
definition of the connectivity of a surface, the result of the proposition would be taken as the
definition : and the resolution of the surface into one, which is simply connected, would then be
obtained by developing the preceding theory in the reverse order.

332

DEFORMATION

[168.

drawn. Thus in the figure Cl and C2 will form a complete system. The circuits C3 and (74
are also irreducible : they can evidently be deformed into C^ and <72 and reducible circuits
by continuous deformation : in the algebraical notation adopted, we have

C3=C1 + C2, Ci=Cl-C.2.

Ex. 3. Another example of a triply connected surface is given in Fig. 47. Two irredu
cible simple circuits are Cv and C%. Another irreducible circuit is C3; this can be

Fig. 47.

reconciled with Cl and C.2 by drawing the point a into coincidence with the intersection
of Cj and (72, and the point c into coincidence with the same point.

Ex. 4. As a last example, consider the surface of a solid sphere with n holes bored
through it. The connectivity is 2n + 1 : hence 2n independent irreducible simple circuits

Fig. 48.

can be drawn on the surface. The simplest complete system is obtained by taking 2n
curves : made up of a set of n, each round one hole, and another set of n, each through
one hole.

A resolution of this surface is given by taking cross-cuts, one round each hole (making
the circuits through the holes no longer possible) and one through each hole (making the
circuits round the holes no longer possible).

The simplest case is that for which n= 1 : the surface is equivalent to the anchor-ring.

169. Surfaces are at present being considered in view of their use as a
means of representing the value of a complex variable. The foregoing inves
tigations imply that surfaces can be classed according to their connectivity ;
and thus, having regard to their designed use, the question arises as to
whether all surfaces of the same connectivity arc equivalent to one another,
so as to be transformable into one another.

169.] OF CONNECTED SURFACES 333

Moreover, a surface can be physically deformed and still remain suitable for
representation of the variable, provided certain conditions are satisfied. We
thus consider geometrical transformation as well as physical deformation ; but
we are dealing only with the general results and not with the mathematical
relations of stretching and bending, which are discussed in treatises on
Analytical Geometry*.

It is evident that continuity is necessary for both : discontinuity would
imply discontinuity in the representation of the variable. Points that are
contiguous (that is, separated only by small distances measured in the surface)
must remain contiguous -f*: and one point in the unchanged surface must
correspond to only one point in the changed surface. Hence in the continuous
deformation of a surface there may be stretching and there may be bending ;
but there must be no tearing and there must be no joining.

For instance, a single untwisted ribbon, if cut, comes to be simply connected. If a twist
through 180° be then given to one end and that end be then joined to the other, we shall
have a once- twisted ribbon, which is a surface with only one face and only one edge;
it cannot be looked upon as an equivalent of the former surface.

A spherical surface with a single hole can have the hole stretched and the surface
flattened, so as to be the same as a bounded portion of a plane : the two surfaces are
equivalent to one another. Again, in the spherical surface, let a large indentation be
made : let both the outer and the inner surfaces be made spherical ; and let the mouth of
the indentation be contracted into the form of a long, narrow hole along a part of a great
circle. When each point of the inner surface is geometrically moved so that it occupies the
position of its reflexion in the diametral plane of the hole, the final form§ of the whole
surface is that of a two-sheeted surface with a junction along a line : it is a spherical
winding-surface, and is equivalent to the simply connected spherical surface.

170. It is sufficient, for the purpose of representation, that the two
surfaces should have a point-to-point transformation : it is not necessary
that physical deformation, without tears or joins, should be actually possible.
Thus a ribbon with an even number of twists would be as effective as a
limited portion of a cylinder, or (what is the same thing) an untwisted ribbon :
but it is not possible to deform the one into the other physically |.

It is easy to see that either deformation or transformation of the kind
considered will change a bifacial surface into a bifacial surface ; that it will
not alter the connectivity, for it will not change irreducible circuits into

* See, for instance, Frost's Solid Geometry, (3rd ed.), pp. 342 — 352.

t Distances between points must be measured along the surface, not through space ; the
distance between two points is a length which one point would traverse before reaching the
position of the other, the motion of the point being restricted to take place in the surface.
Examples will arise later, in Biemann's surfaces, in which points that are contiguous in space
are separated by finite distances on the surface.

§ Clifford, Coll. Hath. Papers, p. 250.

J The difference between the two cases is that, in physical deformation, the surfaces are the
surfaces of continuous matter and are impenetrable ; while, in geometrical transformation, the
surfaces may be regarded as penetrable without interference with the continuity.

334 DEFORMATION OF SURFACES [170.

reducible circuits, and the number of independent irreducible circuits
determines the connectivity: and that it will not alter the number of boundary
curves, for a boundary will be changed into a boundary. These are necessary
relations between the two forms of the surface : it is not difficult to see that
they are sufficient for correspondence. For if, on each of two bifacial surfaces
with the same number of boundaries and of the same connectivity, a complete
system of simple irreducible circuits be drawn, then, when the members of the
systems are made to correspond in pairs, the full transformation can be effected
by continuous deformation of those corresponding irreducible circuits. It
therefore follows that : —

The necessary and sufficient conditions, that two bifacial surfaces may be
equivalent to one another for the representation of a variable, are that tlie two
surfaces should be of the same connectivity and should have the same number of
boundaries.

As already indicated, this equivalence is a geometrical equivalence :
deformation may be (but is not of necessity) physically possible.

Similarly, the presence of one or of several knots in a surface makes no
essential difference in the use of the surface for representing a variable. Thus
a long cylindrical surface is changed into an anchor-ring when its ends are
joined together ; but the changed surface would be equally effective for
purposes of representation if a knot were tied in the cylindrical surface before
the ends are joined.

But it need hardly be pointed out that though surfaces, thus twisted or
knotted, are equivalent for the purpose indicated, they are not equivalent for
all topological enumerations.

Seeing that bifacial surfaces, with the same connectivity and the same
number of boundaries, are equivalent to one another, it is natural to adopt, as
the surface of reference, some simple surface with those characteristics; thus
for a surface of connectivity 2p + 1 with a single boundary, the surface of a
solid sphere, bounded by a point and pierced through with p holes, could be
adopted.

Klein calls* such a surface of reference a Normal Surface.

It has been seen that a bounded spherical surface and a bounded simply connected
part of a plane are equivalent — they are, moreover, physically deformable into one
another.

An untwisted closed ribbon is equivalent to a bounded piece of a plane with one hole
in it — they are deformable into one another : but if the ribbon, previous to being closed,
have undergone an even number of twists each through 180°, they are still equivalent
but are not physically deformable into one another. Each of the bifacial surfaces is
doubly connected (for a single cross-cut renders each simply connected) and each of them

* Ueber Riemann's Theorie der algebraischen Functionen und ihrer Integrate, (Leipzig,
Teubner, 1882), p. 26.

170.] REFERENCES 335

has two boundaries. If however the ribbon, previous to being closed, have imdcrgone
an odd number of twists each through 180°, the surface thus obtained is not equivalent to
the single-holed portion of the plane ; it is unifacial arid has only one boundary.

A spherical surface pierced in n-\-l holes is equivalent to a bounded portion of the
plane with n holes ; each is of connectivity n + 1 and has n + 1 boundaries. The spherical
surface can be deformed into the plane surface by stretching one of its holes into the form
of the outside boundary of the plane surface.

Ex. Prove that the surface of a bounded anchor-ring can be physically deformed into
the surface in Fig. 47, p. 332.

For continuation and fuller development of the subjects of the present chapter, the
following references, in addition to those which have been given, will be found useful :

Klein, Math. Ann., t. vii, (1874), pp. 548—557; ib., t. ix, (1876), pp. 476—482.

Lippich, Math. Ann., i. vii, (1874), pp. 212 — 229 ; Wiener Sitzungsb., t. Ixix, (ii),
(1874), pp. 91—99.

Durege, Wiener Sitzungsb., t. Ixix, (ii), (1874), pp. 115—120; and section 9 of his
treatise, quoted on p. 316, note.

Neumann, chapter vii of his treatise, quoted on p. 5, note.

Dyck, Math. Ann., t. xxxii, (1888), pp. 457—512, ib., t. xxxvii, (1890), pp. 273—316;

at the beginning of the first part of this investigation, a valuable series of references
is given.

Dingeldey, Topologische Studien, (Leipzig, Teubner, 1890).

CHAPTER XV.

RIEMANN'S SURFACES.

171. THE method of representing a variable by assigning to it a position
in a plane or on a sphere is effective when properties of uniform functions of
that variable are discussed. But when multiform functions, or integrals of
uniform functions occur, the method is effective only when certain parts of
the plane are excluded, due account being subsequently taken of the effect of
such exclusions; and this process, the extension of Cauchy's method, was
adopted in Chapter IX.

There is another method, referred to in § 100 as due to Riemann, of an
entirely different character. In Riemann's representation, the region, in
which the variable z exists, no longer consists of a single plane but of a
number of planes ; they are distinct from one another in geometrical concep
tion, yet, in order to preserve a representation in which the value of the
variable is obvious on inspection, the planes are infinitesimally close to one
another. The number of planes, often called sheets, is the same as the
number of distinct values (or branches) of the function w for a general
argument z and, unless otherwise stated, will be assumed finite; each sheet
is associated with one branch of the function, and changes from one branch
of the function to another are effected by making the ^-variable change
from one sheet to another, so that, to secure the possibility of change
of sheet, it is necessary to have means of passage from one sheet to another.
The aggregate of all the sheets is a surface, often called a Riemanns
Surface.

For example, consider the function

w=z* + (z-I}~*,

the cube roots being independent of one another. It is evidently a nine-valued function ;
the number of sheets in the appropriate Eiemann's surface is therefore nine.

The branch-points are 2 = 0, z = l, 2=00. Let o> and a denote a cube-root of unity,
independently of one another ; then the values of z* can be represented in the form

171.]

EXAMPLES OF RIEMANN's SURFACES

337

ill -A - 4

23, C023", co22*; and the values of (2-!) 3 can be represented in the form (2-!) ,

^•(z - \ ) ~ 3} 0 (« - 1) » The nine values of w can be symbolically expressed as follows : —

Fig. 49.

Fig. 50.

where the symbols opposite to w give the coefficients of z3 and of (2- 1) 3 respectively.

Now when 2 describes a small simple circuit positively round the origin, the groups
in cyclical order are u\, w2, w3; w4, w5, w6; wr, w8, io9. And therefore, in the immediate
vicinity of the origin, there must be means of passage to enable
the 2-point to make the corresponding changes from sheet to —
sheet. Taking a section of the whole surface near the origin ~
so as to indicate the passages and regarding the right-hand
sides as the part from which the 2-variable moves when it —
describes a circuit positively, the passages must be in character as
indicated in Fig. 49. And it is evident that the further descrip
tion of small simple circuits round the origin will, with these passages, lead to the proper
values : thus %, which after the single description is the value of w4, becomes w6 after
another description and it is evident that a point in the w-0 sheet passes into the w6 sheet.

When 2 describes a small simple circuit positively round the point 1, the groups in cyclical
order are wlt ^4, %; w2, w5, ws; w3, w6, w9: and therefore,
in the immediate vicinity of the point 1, there must be ~
means of passage to render possible the corresponding changes
of 2 from sheet to sheet. Taking a section as before near the ~
point 1 and with similar convention as to the positive direc
tion of the 2-path, the passages must be in character as
indicated in Fig. 50.

Similarly for infinitely large values of 2.

If then the sheets can be so joined as to give these possibilities of passage and also
give combinations of them corresponding to combinations of the simple paths indicated,
then there will be a surface to any point of which will correspond one and only one value
of w : and when the value of w is given for a point 2 in an ordinary plane of variation,
then that value of w will determine the sheet of the surface in which the point 2 is to
be taken. A surface will then have been constructed such that the function w, which is
multiform for the single-plane representation of the variable, is uniform for variations
in the many-sheeted surface.

Again, for the simple example arising from the two-valued function, defined by
the equation

w = {(z-a}(z-b}(z-c}}-\

the branch-points are a, b, c, oo ; and a small simple circuit round any one of these
four points interchanges the two values. The Riemann's surface is two-sheeted and
there must be means of passage between the two sheets in the vicinity of a, that of b,
that of c and at the infinite part of the plane.

These examples are sufficient to indicate the main problem. It is the
construction of a surface in which the independent variable can move so
F. 22

338 SHEETS OF HIEMANN'S SURFACE [171.

that, for variations of z in that surface, the multiformity of the function is
changed to uniformity. From the nature of the case, the character of the
surface will depend on the character of the function : and thus, though all the
functions are uniform within their appropriate surfaces, these surfaces are
widely various. Evidently for uniform functions of z the appropriate surface
on the above method is the single plane already adopted.

172. The simplest classes of functions for which a Riemaim's surface is
useful are (i) those called (§ 94) algebraic functions, that is, multiform functions
of the independent variable denned by an algebraical equation of the form

which is of finite degree, say n, in w, and (ii) those usually called Abelian
functions, which arise through integrals connected with algebraic functions.

Of such an algebraic function there are, in general, n distinct values ; but
for the special values of z, that are the branch-points, two or more of the
values coincide. The appropriate Riemann's surface is composed of n sheets ;
one branch, and only one branch, of w is associated with a sheet. The
variable z, in its relation to the function, is determined not merely by its
modulus and argument but also by its sheet ; that is, in the language of the
earlier method, we take account of the path by which z acquires a value. The
particular sheet in which z lies determines the particular branch of the
function. Variations of #, which occur within a sheet and do not coincide
with points lying in regions of passage between the sheets, lead to variations
in the value of the branch of w associated with the sheet ; a return to an
initial value of z, by a path that nowhere lies within a region of passage,
leaves the ^-point in the same sheet as at first and so leads to the initial
branch (and to the initial value of the branch) of w. But a return to an
initial value of z by a path, which, in the former method of representation,
would enclose a branch-point, implies a change of the branch of the function
according to the definite order prescribed by the branch-point. Hence the
final value of the variable z on the Riemann's surface must lie in a sheet that
is different from that of the initial (and algebraically equal) value ; and
therefore the sheets must be so connected that, in the immediate vicinity of
branch-points, there are means of passage from one sheet to another, securing
the proper interchanges of the branches of the function as defined by the
equation.

173. The first necessity is therefore the consideration of the mode in
which the sheets of a Riemann's surface are joined : the mode is indicated by
the theorem that sheets of a Riemann's surface are joined along lines.

The junction might be made either at a point, as with two spheres in
contact, or by a common portion of a surface, as with one prism lying on

173.] JOINED ALONG BRANCH-LINES 339

another, or along lines ; but whatever the character of the junction be, it
must be such that a single passage across it (thereby implying entrance to
the junction and exit from it) must change the sheet of the variable.

If the junction were at a point, then the £- variable could change from one
sheet into another sheet, only if its path passed through that point : any
other closed path would leave the z- variable in its original sheet. A small
closed curve, infinites! rn ally near the point and enclosing it and no other
branch-point, is one which ought to transfer the variable to another sheet
because it encloses a branch-point : and this is impossible with a point-junction
when the path does not pass through the point. Hence a junction at a point
only is insufficient to provide the proper means of passage from sheet to
sheet.

If the junction were effected by a common portion
of surface, then a passage through it (implying an
entrance into that portion and an exit from it) ought to
change the sheet. But, in such a case, closed contours .-'--''
can be constructed which make such a passage without Fi8- 51>

enclosing the branch-point a : thus the junction would cause a change of
sheet for certain circuits the description of which ought to leave the
z- variable in the original sheet. Hence a junction by a continuous area of
surface does not provide proper means of passage from sheet to sheet.

The only possible junction which remains is a line.

The objection in the last case does not apply to a closed • / '^

contour which does not contain the branch-point ; for the /.--"''

line cuts the curve twice and there are therefore two Fig. 52.

crossings ; the second of them makes the variable return to the sheet which
the first crossing compelled it to leave.

Hence the junction between any two sheets takes place along a line.

Such a line is called* a branch-line. The branch -points of a multiform
function lie on the branch-lines, after the foregoing explanations ; and a
branch-line can be crossed by the variable only if the variable change its
sheet at crossing, in the sequence prescribed by the branch-point of the
function which lies on the line. Also, the sequence is reversed when the
branch-line is crossed in the reversed direction.

Thus, if two sheets of a surface be connected along a branch-line, a point which
crosses the line from the first sheet must pass into the second and a point which crosses
the line from the second sheet must pass into the first.

Again, if, along a common direction of branch-line, the first sheet of a surface
be connected with the second, the second with the third, and the third with

* Sometimes cross-line, sometimes branch-section. The German title is Verzweigungschnitt;
the French is lignc de passage ; see also the note on the equivalents of branch-point, p. 15.

22—2

340 PROPERTIES OF BRANCH-LINES [173.

the first, a point which crosses the line from the first sheet in one direction must pass
into the second sheet, but if it cross the line in the other direction it must pass into
the third sheet.

A branch -point does not necessarily affect all the branches of a function :
when it affects only some of them, the corresponding property of the Riemann's
surface is in evidence as follows. Let z=a determine a branch-point affecting,
say, only r branches. Take n points a, one in each of the sheets ; and through
them draw n lines cab, having the same geometrical position in the respective
sheets. Then in the vicinity of the point a in each of the n sheets, associated
with the r affected branches, there must be means of passage from each one
to all the rest of them ; and the lines cab can conceivably be the branch-lines
with a properly established sequence. The point a does not affect the other
n — r branches : there is therefore no necessity for means of passage in the
vicinity of a among the remaining n — r sheets. In each of these remaining
sheets, the point a and the line cab belong to their respective sheets alone :
for them, the point a is not a branch-point and the line cab is not a branch-
line.

174. Several essential properties of the branch-lines are immediate
inferences from these conditions.

I. A free end of a branch-line in a surface is a branch-point.

Let a simple circuit be drawn round the free end so small as to enclose no
branch-point (except the free end, if it be a branch-point). The circuit meets
the branch-line once, and the sheet is changed because the branch-line is
crossed ; hence the circuit includes a branch-point which therefore can be
only the free end of the line.

Note. A branch-line may terminate in the boundary of the surface,
and then the extremity need not be a branch-point.

II. When a branch-line extends beyond a branch-point lying in its course,
the sequence of interchange is not the same on the two sides of the point.

If the sequence of interchange be the same on the two sides of the branch
point, a small circuit round the point would first cross one part of the branch-
line and therefore involve a change of sheet and then, in its course, would
cross the other part of the branch-line in the other direction which, on the
supposition of unaltered sequence, would cause a return to the initial sheet.
In that case, a circuit round the branch-point would fail to secure the proper
change of sheet. Hence the sequence of interchange caused by the branch-
line cannot be the same on the two sides of the point.

III. If two branch-lines with different sequences of interchange have a
common extremity, that point is either a branch-point or an extremity of at
least one other branch-line.

174.] SYSTEM OF BRANCH-LINES 341

If the point be not a branch-point, then a simple curve enclosing it, taken
so small as to include no branch-point, must
leave the variable in its initial sheet. Let A
be such a point, AB and AC be two branch-
lines having A for a common extremity ; let ., A ,.• — ^ «

the sequence be as in the figure, taken for a F.

simple case ; and suppose that the variable

initially is in the rth sheet. A passage across AB makes the variable
pass into the sth sheet. If there be no branch-line between AB and AC
having an extremity at A, and if neither n nor m be s, then the passage
across AC makes no change in the sheet of the variable and, therefore, in
order to restore r before AB, at least one branch-line must lie in the angle
between AC and AB, estimated in the positive trigonometrical sense.

If either n or m, say n, be s, then after passage across AC, the point is in
the mt\i sheet ; then, since the sequences are not the same, m is not r and
there must be some branch-line between AC and AB to make the point
return to the rth sheet on the completion of the circuit.

If then the point A be not a branch-point, there must be at least one
other branch-line having its extremity at A. This proves the proposition.

COROLLARY 1. If both of two branch-lines extend beyond a point of inter
section, which is not a branch-point, no sheet of the surface has both of them for
branch-lines.

COROLLARY 2. If a change of sequence occur at any point of a branch-
line, then either that point is a branch-point or it lies also on some other
branch-line.

COROLLARY 3. No part of a branch-line with only one branch-point on it
can be a closed curve.

It is evidently superfluous to have a branch-line without any branch-point
on it.

175. On the basis of these properties, we can obtain a system of branch-
lines satisfying the requisite conditions which are : —

(i) the proper sequences of change from sheet to sheet must be
secured by a description of a simple circuit round a branch
point : if this be satisfied for each of the branch-points, it
will evidently be satisfied for any combination of simple circuits,
that is, for any path whatever enclosing one or more branch
points.

(ii) the sheet, in which the variable re-assumes its initial value after
describing a circuit that encloses no branch-point, must be the
initial sheet.

342 SYSTEM OF BRANCH-LINES [175.

In the ^-plane of Cauchy's method, let lines be drawn from any point I, not
a branch-point in the first instance, to each of the branch-points, as in fig. 19,
p. 156, so that the joining lines do not meet except at /: and suppose the
w-sheeted Riemann's surface to have branch-lines coinciding geometrically
with these lines, as in § 173, and having the sequence of interchange for
passage across each the same as the order in the cycle of functional values
for a small circuit round the branch-point at its free end. No line (or part
of a line) can be a closed curve ; the lines need not be straight, but they
will be supposed drawn as direct as possible to the points in angular
succession.

The first of the above requisite conditions is satisfied by the establish
ment of the sequence of interchange.

To consider the second of the conditions, it is convenient to divide
circuits into two kinds, (a) those which exclude /, (/3) those which include /,
no one of either kind (for our present purpose) including a branch-point.

A closed circuit, excluding I and all the branch-points, must intersect a
branch-line an even number of times, if
it intersect the line in real points. Let
the figure (fig. 54) represent such a case :
then the crossings at A and B counter
act one another and so the part be
tween A and B may without effect be
transferred across IB3 so as not to cut
the branch-line at all. Similarly for
the points C and D : and a similar
transference of the part now between
C and D may be made across the
branch-line without effect: that is, the
circuit can, without effect, be changed
so as not to cut the branch-line IBS at all. A similar change can be made
for each of the branch-lines : and so the circuit can, without effect, be changed
into one which meets no branch-line and therefore, on its completion, leaves
the sheet unchanged.

A closed circuit, including / but no branch-point, must meet each branch-
line an odd number of times. A change similar in character to that in
the previous case may be made for each branch-line : and without affecting
the result, the circuit can be changed so that it meets each branch-line only
once. Now the effect produced by a branch-line on the function is the same
as the description of a simple loop round the branch-point which with /
determines the branch-line : and therefore the effect of the circuit at present
contemplated is, after the transformation which does not affect the result, the
same as that of a circuit, in the previously adopted mode of representation,

175.] FOR A SURFACE

enclosing all the branch-points. But, by Cor. III. of § 90, the effect of a
circuit which encloses all the branch-points (including z = GO , if it be a
branch-point) is to restore the value of the function which it had at the
beginning of the circuit : and therefore in the present case the effect is to
make the point return to the sheet in which it lay initially.

It follows therefore that, for both kinds of a closed circuit containing no
branch-point, the effect is to make the ^-variable return to its initial sheet
on resuming its initial value at the close of the circuit.

Next, let the point / be a branch-point ; and let it be joined by lines,
as direct* as possible, to each of the other branch -points in angular succes
sion. These lines will be regarded as the branch-lines ; and the sequence of
interchange for passage across any one is made that of the interchange pre
scribed by the branch-point at its free extremity.

The proper sequence of change is secured for a description of a simple
closed circuit round each of the branch-points other than /. Let a small
circuit be described round /; it meets each of the branch-lines once and
therefore its effect is the same as, in the language of the earlier method of
representing variation of z, that of a circuit enclosing all the branch-points
except 7. Such a circuit, when taken on the Neumann's sphere, as in Cor.
III., § 90 and Ex. 2, § 104, may be regarded in two ways, according as one or
other of the portions, into which it divides the area of the sphere, is regarded
as the included area; in one way, it is a circuit enclosing all the branch
points except /, in the other it is a circuit enclosing / alone and no other
branch-point. Without making any modification in the final value of w, it
can (by § 90) be deformed, either into a succession of loops round all the
branch-points save one, or into a loop round that one ; the effect of these two
deformations is therefore the same. Hence the effect of the small closed
circuit round / meeting all the branch-lines is the same as, in the other mode
of representation, that of a small curve round / enclosing no other branch
point ; and therefore the adopted set of branch- lines secures the proper
sequence of change of value for description of a circuit round /.

The first of the two necessary conditions is therefore satisfied by the
present arrangement of branch-lines.

The proof, that the second of the two necessary conditions is also satisfied
by the present arrangement of branch-lines, is similar to that in the preceding
case, save that only the first kind of circuit of the earlier proof is possible.

Jt thus appears that a system of branch-lines can be obtained which
secures the proper changes of sheet for a multiform function : and therefore
Riemann's surfaces can be constructed for such a function, the essential
property being that over its appropriate surface an otherwise multiform
function of the variable is a uniform function.

* The reason for this will appear in §§ 183, 184.

344 EXAMPLES [175.

The multipartite character of the function has its influence preserved by
the character of the surface to which the function is referred : the surface,
consisting of a number of sheets joined to one another, may be a multiply
connected surface.

In thus proving the general existence of appropriate surfaces, there has
remained a large arbitrary element in their actual construction : moreover,
in particular cases, there are methods of obtaining varied configurations of
branch-lines. Thus the assignment of the n branches to the n sheets has
been left unspecified, and is therefore so far arbitrary : the point I, if not a
branch-point, is arbitrarily chosen and so there is a certain arbitrariness of
position in the branch -lines. Naturally, what is desired is the simplest
appropriate surface : the particularisation of the preceding arbitrary qualities
is used to derive a canonical form of the surface.

176. The discussion of one or two simple cases will help to illustrate the
mode of junction between the sheets, made by branch-lines.

The simplest case of all is that in which the surface has only a single
sheet: it does not require discussion.

The case next in simplicity is that in which the surface is two-sheeted :
the function is therefore two- valued and is consequently defined by a
quadratic equation of the form

Lua + 2Mu + N = 0,

where L and M are uniform functions of z. When a new variable w is
introduced, defined by Lu + M=w, so that values of iv and of u correspond
uniquely, the equation is

It is evident that every branch-point of u is a branch-point of w, and
vice versa ; hence the Riemann's surface is the same for the two equations.
Now any root of P (z) of odd degree is a branch-point of iv. If then

where R (z} is a product of only simple factors, every factor of R (z) leads to
a branch-point. If the degree of R (z} be even, the number of branch-points
for finite values of the variable is even and z = oo is not a branch-point ; if the
degree of R(z) be odd, the number of branch -points for finite values of the
variable is odd and z = oo is a branch-point : in either case, the number of
branch-points is even.

There are only two values of w, and the Riemann's surface is two-sheeted:
crossing a branch-line therefore merely causes a change of sheet. The free
ends of branch-lines are branch-points ; a small circuit round any branch
point causes an interchange of the branches w, and a circuit round any two
branch-points restores the initial value of w at the end and therefore leaves
the variable in the same sheet as at the beginning. These are the essential
requirements in the present case ; all of them are satisfied by taking each

176.] OF RIEMANN'S SURFACES 345

branch-line as a line connecting two (and only two) of the branch-points. The
ends of all the branch -lines are free : and their number, in this method, is
one-half that of the (even) number of branch-points. A small circuit round
a branch-point meets a branch-line once and causes a change of sheet ; a
circuit round two (and not more than two) branch -points causes either no
crossing of branch-line or an even number of crossings and therefore restores
the variable to the initial sheet.

A branch-line is, in this case, usually drawn in the form of a straight line
when the surface is plane : but this form is not essential and all that is
desirable is to prevent intersections of the branch-lines.

Note. Junction between the sheets along a branch-line is easily secured.
The two sheets to be joined are cut along the branch-line. One edge of the
cut in the upper sheet, say its right edge looking along the section, is joined
to the left edge of the cut in the lower sheet ; and the left edge in the upper
sheet is joined to the right edge in the lower.

A few simple examples will illustrate these remarks as to the sheets : illustrations of
closed circuits will arise later, in the consideration of integrals of multiform functions.

Ex. 1. Let w* = A(z-a)(z-b},

so that a and b are the only branch-points. The surface is two-sheeted : the line ab may
be made the branch-line. In Fig. 55 only part of the upper sheet is shewn*, as likewise
only part of the lower sheet. Continuous lines imply what is visible ; arid dotted lines
what is invisible, on the supposition that the sheets are opaque.

The circuit, closed in the surface and passing round 0, is made up of OJK in the upper
sheet : the point crosses the branch-line and then passes into the lower sheet, where it
describes the dotted line KLH : it then meets and crosses the branch-line at If, passes
into the upper sheet and in that sheet returns to 0. Similarly of the line ABC, the part
AB lies in the lower sheet, the part EC in the upper : of the line DG the part DE lies in
the upper sheet, the part EFG in the lower, the piece FG of this part being there visible
beyond the boundary of the retained portion of the upper surface.

Ex. 2. Let Aw?2 = z3-a3.

The branch-points (Fig. 56) are A ( = a), B ( = ««), (7( = aa2), where a is a primitive cube
root of unity, and 2 = 00. The branch -lines can be made by BC, Ace ; and the two-
sheeted surface will be a surface over which w is uniform. Only a part of each sheet
is shewn in the figure; a section also is made at M across the surface, cutting the branch -
line A QO .

Ex. 3. Let wm=zn,

where n and TO are prime to each other. The branch-points are z = 0 and 2=00 ; and the
branch-line extends from 0 to QO . There are m sheets ; if we associate them in order with
the branches ws, where

wa=re

for s=l, 2, ..., TO, then the first sheet is connected with the second forwards, the second
with the third forwards, and so on ; the mth being connected with the first forwards.

* The form of the three figures in the plate opposite p. 346 is suggested by Holzmiiller, Ein-
fiihrung in die Theorie der isogonalen Vericandschaften und der confomien AbbUdimgen, (Leipzig,
Teubner, 1882), in which several illustrations are given.

346

SPHERICAL RIEMANN'S SURFACE

[176.

The surface is sometimes also called a winding-surface; and a branch-point such as
z—0 on the surface, where a number m of sheets pass into one another in succession, is
also called a winding-point of order m— 1 (see p. 15, note). An illustration of the surface
for m = 3 is given in Fig. 57, the branch-line being cut so as to shew the branching : what
is visible is indicated by continuous lines ; what is in the second sheet, but is invisible, is
indicated by the thickly dotted line ; what is in the third sheet, but is invisible, is indic
ated by the thinly dotted line.

Ex. 4. Consider a three-sheeted surface having four branch-points at a, b, c, d ; and
let each point interchange two branches, say, w.2, w3 at a ; iv^ w3ai b ; w2, w3 at c ; wlt w2

at d ; the points being as in Fig. 58. It is easy to verify that these branch-points
satisfy the condition that a circuit, enclosing them all, restores the initial value of w.

The branching of the sheets may be made as in the figure, the integers on the two sides
of the line indicating the sheets that are to be joined along the line.

A canonical form for such a surface can be derived from the more general case given
later (in §§ 186—189).

Ex. 5. Shew that, if the equation

be of degree n in w and be irreducible, all the n sheets of the surface are connected, that
is, it is possible by an appropriate path to pass from any sheet to any other sheet.

177. It is not necessary to limit the surface representing the variable to
a set of planes; and, indeed, as with uniform functions, there is a convenience
in using the sphere for the purpose.

We take n spheres, each of diameter unity, touching the Riemann's plane
surface at a point A ; each sphere is regarded as the stereographic projection
of a plane sheet, with regard to the other extremity A' of the spherical
diameter through A. Then, the sequence of these spherical sheets being
the same as the sequence of the plane sheets, branch-points in the plane
surface project into branch-points on the spherical surface : branch -lines be
tween the plane sheets project into branch-lines between the spherical sheets
and are terminated by corresponding points ; and if a branch-line extend in
the plane surface to z=co, the corresponding branch-line in the spherical
surface is terminated at A'.

A surface will thus be obtained consisting of n spherical sheets; like
the plane Riemann's surface, it is one over which the n-valued function is a
uniform function of the position of the variable point.

Fig.

M — =-00

To face p. 346.

Fig. 57.

177.] CONNECTIVITY OF A RIEMANN's SURFACE 347

But also the connectivity of the n-sheeted spherical surface is the same as
that of the n-sheeted plane surface with which it is associated.

In fact, the plane surface can be mechanically changed into the spherical
surface without tearing, or repairing, or any change except bending and
compression: all that needs to be done is that the n plane sheets shall be
bent, without making any change in their sequence, each into a spherical
form, and that the boundaries at infinity (if any) in the plane-sheet shall
be compressed into an infinitesimal point, being the South pole of the cor
responding spherical sheet or sheets. Any junctions between the plane
sheets extending to infinity are junctions terminated at the South pole. As
the plane surface has a boundary, which, if at infinity on one of the sheets, is
therefore not a branch-line for that sheet, so the spherical surface has a
boundary which, if at the South pole, cannot be the extremity of a branch-
line.

178. We proceed to obtain the connectivity of a Riemann's surface : it
is determined by the following theorem : —

Let the total number of branch-points in a Riemann's n-sheeted surface be
r ; and let the number of, branches of the function interchanging at the first
point be ml, the number interchanging at the second be m.2, and so on. Then
the connectivity of the surface is

fl-2n + 3,
where fl denotes m,1 + m2 + ... + mr — r.

Take* the surface in the bounded spherical form, the connectivity N of
which is the same as that of the plane surface : and let the boundary be a
small hole A in the outer sheet. By means of cross-cuts and loop-cuts, the
surface can be resolved into a number of distinct simply connected pieces.

First, make a slice bodily through the sphere, the edge in the
outside sheet meeting A and the direction of the
slice through A being chosen so that none of the
branch -points lie in any of the pieces cut off. Then n
parts, one from each sheet and each simply connected,
are taken away. The remainder of the surface has a
cup-like form ; let the connectivity of this remainder
be M.

This slice has implied a number of cuts.

The cut made in the outside sheet is a cross-cut,
because it begins and ends in the boundary A. It
divides the surface into two distinct pieces, one being
the portion of the outside sheet cut off, and this piece is simply connected ;

* The proof is founded on Neumann's, pp. 108 — 172.

348 CONNECTIVITY OF A SURFACE [178.

hence, by Prop. III. of § 160, the remainder has its connectivity still repre
sented by N.

The cuts in all the other sheets, caused by the slice, are all loop-cuts,
because they do not anywhere meet the boundary. There are n — 1 loop-
cuts, and each cuts off a simply connected piece ; and the remaining surface
is of connectivity M. Hence, by Prop. V. of § 161,

M + n - 1 = N + 2 (n - 1),
and therefore M = N+n—l.

In this remainder, of connectivity M, make r — 1 cuts, each of which
begins in the rim and returns to the rim, and is to be made through the n
sheets together ; and choose the directions of these cuts so that each of the
r resulting portions of the surface contains one (and only one) of the branch
points.

Consider the portion of the surface which contains the branch-point
where ml sheets of the surface are connected. The ml connected sheets
constitute a piece of a winding-surface round the winding-point of order
ml — 1 ; the remaining sheets are unaffected by the winding-point, and
therefore the parts of them are n — m^ distinct simply connected pieces.
The piece of winding- surface is simply connected ; because a circuit, that
does not contain the winding-point, is reducible without passing over the
winding-point, and a circuit, that does contain the winding-point, is reducible
to the winding-point, so that no irreducible circuit can be drawn. Hence
the portion of the surface under consideration consists of n — ml + 1 distinct
simply connected pieces.

Similarly for the other portions. Hence the total number of distinct
simply connected pieces is

r

2 (n - mq + 1)

9 = 1

r

= m — 2 mq + r
l-i

= nr — fl.

But in the portion of connectivity M each of the r — 1 cuts causes, in
each of the sheets, a cut passing from the boundary and returning to the
boundary, that is, a cross-cut. Hence there are n cross-cuts from each of the
r—\ cuts, and therefore n (r— 1) cross-cuts altogether, made in the portion of
surface of connectivity M.

The effect of these n(r — 1) cross-cuts is to resolve the portion of con
nectivity M into nr — £l distinct simply connected pieces ; hence, by § 160,

M = n (r - 1) - (nr - H) + 2,

and therefore N = M — (n — 1) = n - 2n + 3,

the connectivity of the Riemann's surface.

178.] CLASS OF A SURFACE 349

r

The quantity H, having the value 2 (mq — 1), may be called the rami-

</=i

fication of the surface, as indicating the aggregate sum of the orders of
the different branch-points.

Note. The surface just considered is a closed surface to which a point
has been assigned for boundary; hence, by Cor. I., Prop. III., § 164, its
connectivity is an odd integer. Let it be denoted by 2p + 1 ; then

2p = ft - 2/i + 2,

and 2p is the number of cross-cuts which change the Riemann's surface into
one that is simply connected.

The integer p is often called (Cor. I., Prop. III., § 164) the class of the
Riemann's surface; and the equation

f(w, z) = 0

is said to be of class p, when p is the class of the associated Riemann's
surface.

Ex. 1. When the equation is

w> = \(z-a}(z-b\

we have a two-sheeted surface, ?t = 2. There are two branch- points, z = a and z = b; but
2=00 is not a branch-point ; so that r=2. At each of the branch-points the two values are
interchanged, so that m1 = 2, ??i2 = 2; thus Q = 2. Hence the connectivity =2-4 + 3 = 1,
that is, the surface is simply connected.

The surface can be deformed, as in the example in § 169, into a sphere.
Ex. 2. When the equation is

we have ?t = 2. There are four branch-points, viz., et, e2, e3, oc , so that r = 4 ; and at each
of them the two values of w are interchanged, so that mg = 2 (for 5 = 1,2, 3, 4), and therefore
Q = 8- 4 = 4. Hence the connectivity is 4- 4 + 3, that is, 3 ; and the value of p is unity.

Similarly, the surface associated with the equation

where U(z] is a rational, integral, algebraical function of degree 2«i - 1 or of degree 2»i,
is of connectivity 2wi + l ; so that p = m. The equation

W2==(1_22)(1_^2)

is of class p=\. The case next in importance is that of the algebraical equation leading to
the hyperelliptic functions, when (/"is either a quintic or a sextic ; and then p = 2.

Ex. 3. Obtain the connectivity of the Riemann's surface associated with the equation

w3 + ^ — 3awz = 1 ,
where a is a constant, (i) when a is zero, (ii) when a is different from zero.

•350 RESOLUTION OF A RIEMANN's SURFACE [178.

Ex. 4. Shew that, if the surface associated with the equation

f(w,z) = 0,

have p. boundary-lines instead of one, and if the equation have the same branch-points
as in the foregoing proposition, the connectivity is Q-

179. The consideration of irreducible circuits on the surface at once
reveals the multiple connection of the surface, the numerical measure of
which has been obtained. In a Riemann's surface, a simple
closed circuit cannot be deformed over a branch-point. Let
A be a branch-point, and let AE... be the branch-line
having a free end at A. Take a curve ...CED... crossing
the branch-line at E and passing into a sheet different
from that which contains the portion CE ; and, if possible,
let a slight deformation of the curve be made so as to transfer the portion
CE across the branch-point A. In the deformed position, the curve
...C'E'D' '... does not meet the branch-line; there is, consequently, no
change of sheet in its course near A and therefore E'D'..., which is the
continuation of ...C'E', cannot be regarded as the deformed position of ED.
The two paths are essentially distinct ; and thus the original path cannot be
deformed over the branch-point.

It therefore follows that continuous deformation of a circuit over a
branch-point on a Riemann's surface is a geometrical impossibility.

Ex. Trace the variation of the curve CED, as the point E moves up to A and then
returns along the other side of the branch-line.

Hence a circuit containing two or more of the branch-points is irreducible ;
but a circuit containing all the branch-points is equivalent to a circuit that
contains none of them, and it is therefore reducible.

If a circuit contain only one branch-point, it can be continuously deformed
so as to coincide with the point on each sheet and therefore, being deformable
into a point, it is a reducible circuit. An illustration has already occurred in
the case of a portion of winding-surface containing a single winding-point
(p. 348); all circuits drawn on it are reducible.

It follows from the preceding results that the Riemann's surface associated
with a multiform function is generally one of multiple connection ; we shall
find it convenient to know how it can be resolved, by means of cross-cuts, into
a simply connected surface. The representative surface will be supposed a
closed surface with a single boundary ; its connectivity, necessarily odd, being
2/) + l, the number of cross-cuts necessary to resolve the surface into one
that is simply connected is 2p ; when these cuts have been made, the simply
connected surface then obtained will have its boundary composed of a single
closed curve.

179.]

BY CROSS-CUTS

351

One or two simple examples of resolution of special Riemann's surfaces will be useful
in leading up to the general explanation ; in the examples it will be shewn how, in
conformity with § 168, the resolving cross-cuts render irreducible circuits impossible.

Ex. 1. Let *he equation be

w1 = A(z-d)(z-b'](z-c}(z-d\

where a, b, c, d are four distinct points, all of finite modulus. The surface is two-sheeted ;
each of the points a, b, c, d is a branch-point where the two values of w interchange ; and
so the surface, assumed to have a single boundary, is triply connected, the value of p
being unity. The branch-lines are two, each connecting a pair of branch-points ; let them
be ab and cd.

Two cross-cuts are necessary and sufficient to resolve the surface into one that is
simply connected. We first make a cross-cut,
beginning at the boundary S, (say it is in the
upper sheet), continuing in that sheet and re
turning to J3, so that its course encloses the
branch-line ab (but not cd) and meets no branch-
line. It is a cross-cut, and not a loop-cut, for it
begins and ends in the boundary ; it is evidently
a cut in the upper sheet alone, and does not
divide the surface into distinct portions ; and,
once made, it is to be regarded as boundary for
the partially cut surface.

The surface in its present condition is con
nected : and therefore it is possible to pass from one edge to the other of the cut just
made. Let P be a point on it ; a curve that passes from one edge to the other is indicated
by the line PQR in the upper sheet, RS in the lower, and SP in the upper. Along this
line make a cut, beginning at P and returning to P ; it is a cross-cut, partly in the
upper sheet and partly in the lower, and it does not divide the surface into distinct
portions.

Two cross-cuts in the triply connected surface have now been made ; neither of them,
as made, divides the surface into distinct portions, and each of them when made reduces
the connectivity by one unit ; hence the surface is now simply connected. It is easy to
see that the boundary consists of a single line not intersecting itself; for beginning
at P, we have the outer edge of PUT, then the inner edge of 2'QltSP, then the inner
edge of PTB, and then the outer edge of PSRQP, returning to P.

The required resolution has been effected.

Before the surface was resolved, a number of irreducible circuits could be drawn ; a
complete system of irreducible circuits is composed of two, by § 168. Such a system may
be taken in various ways ; let it be composed of a simple curve C lying in the upper sheet
and containing the points a and b, and a simple curve D, lying partly in the upper
and partly in the lower sheet and containing the points a and c ; each of these curves
is irreducible, because it encloses two branch-points. Every other irreducible circuit
is reconcileable with these two ; the actual reconciliation in particular cases is effected
most simply when the surface is taken in a spherical form.

The irreducible circuit C on the unresolved surface is impossible on the resolved
surface owing to the cross-cut SPQRS ; and the irreducible circuit D on the unresolved
surface is impossible on the resolved surface owing to the cross-cut PTB. It is easy
to verify that no irreducible circuit can be drawn on the resolved surface.

352

RESOLUTION

[179.

In practice, it is conveniently effective to select a complete system of irreducible
simple circuits and then to make the cross-cuts so that each of them renders one circuit
of the system impossible on the resolved surface.

Ex. 2. If the equation be

= 4:(z-e1)(z-e.t)(z-e3),

the branch-points are els e2, e3 and oo . When the two-sheeted surface is spherical, and the
branch-lines are taken to be (i) a line joining elf e.2', and (ii) a line joining e3 to the South
pole, the discussion of the surface is similar in detail to that in the preceding example.

Ex. 3. Let the equation be

t*«-4* (!-«)(*-*) <X-*)&*-«),

and for simplicity suppose that AC, X, /* are real quantities subject to the inequalities

The associated surface is two-sheeted and has a boundary assigned to it ; assuming
that its sheets are planes, we shall take some point in the finite part of the upper sheet,
not being a branch-point, as the boundary. There are six . branch-points, viz., 0, 1, K,
X, /x, co at each of which the two values of w interchange ; and so the connectivity of the
surface is 5 and its class, p, is 2. The branch-lines can be taken as three, this being
the simplest arrangement ; let them be the lines joining 0, 1 ; K, X ; /*, oo .

Four cross-cuts are necessary to resolve the surface into one that is simply connected
and has a single boundary. They may be obtained as follows.

Fig. 62.

Beginning at the boundary L, let a cut LHA be made entirely in the upper sheet
along a line which, when complete, encloses the points 0 and 1 but no other branch-points ;
let the cut return to L. This is a cross-cut and it does not divide the surface into
distinct pieces ; hence, after it is made, the connectivity of the modified surface is 4, and
there are two boundary lines, being the two edges of the cut LHA.

Beginning at a point A in LHA, make a cut along ABC in the upper sheet until
it meets the branch-line /zoo, then in the lower sheet along CSD until it meets the
branch-line 01, and then in the upper sheet from D returning to the initial point A.
This is a cross-cut and it does not divide the surface into distinct pieces ; hence, after it
is made, the connectivity of the modified surface is 3, and it is easy to see that there
is only one boundary edge, similar to the single boundary in Ex. 1 when the surface
in that example has been completely resolved.

Make a loop-cut EFG along a line, enclosing the points K and X but no other branch
points ; and change it into a cross-cut by making a cut from E to some point B of the
boundary. This cross-cut can be regarded as BEFGE, ending at a point in its own
earlier course. As it does not divide the surface into distinct pieces, the connectivity is
reduced to 2 ; and there are two boundary lines.

179.] BY CROSS-CUTS 353

Beginning at a point G make another cross-cut GQPRG, as in the figure, enclosing
the two branch-points X and p, and lying partly in the upper sheet and partly in the lower.
It does not divide the surface into distinct pieces : the connectivity is reduced to unity
and there is a single boundary line.

Four cross-cuts have been made ; and the surface has been resolved into one that is
simply connected.

It is easy to verify :

(i) that neither in the upper sheet, nor in the lower sheet, nor partly in the
upper sheet and partly in the lower, can an irreducible circuit be drawn in the resolved
surface ; and

• (ii) that, owing to the cross-cuts, the simplest irreducible circuits in the unresolved
surface — viz. those which enclose 0, 1 ; 1, K ; *, X ; X, /i ; respectively — are rendered
impossible in the resolved surface.

The equation in the present example, and the Riemann's surface associated with it,
lead to the theory of hyperelliptic functions*.

180. The last example suggests a method of resolving any two-sheeted
surface into a surface that is simply connected.

The number of its branch-points is necessarily even, say 2p + 2. The
branch-lines can be made to join these points in pairs, so that there will be
p + l of them. To determine the connectivity (§ 178), we have n = 2 and,
since two values are interchanged at every branch-point, H = 2p -f 2 ; so
that the connectivity is 2p + 1. Then 2p cross-cuts are necessary for the
required resolution of the surface.

We make cuts round p of the branch-lines, that is, round all of them but
one ; each cut is made to enclose two branch-points, and each lies entirely in
the upper sheet. These are cuts corresponding to the cuts LHA and EFG
in fig. 62 ; and, as there, the cut round the first branch-line begins and ends
in the boundary, so that it is a cross-cut. All the remaining cuts are loop-
cuts at present. The system of p cuts we denote by a1} a2, ..., ap.

We make other p cuts, one passing from the inner edge of each of the p
cuts a already made to the branch-line which it surrounds, then in the lower
sheet to the (j) + l)th branch-line, and then in the upper sheet returning to
the point of the outer edge of the cut a at which it began. This system of
cuts corresponds to the cuts ADSGBA and GQPRG in fig. 62. Each of them
can be taken so as to meet no one of the cuts a except the one in which it
begins and ends ; and they can be taken so as not to meet one another.
This system of p cuts we denote by bl} b.2, ..., bp, where br is the cut which
begins and ends in ar. All these cuts are cross-cuts, because they begin and
end in boundary-lines.

Lastly, we make other p — 1 cuts from ar to 6.r_1} for r = 2, 3, . .., p, all in

* One of the most direct discussions of the theory from this point of view is given by Prym,
Neue Theorie der ultraelliptischen Functionen, (Berlin, Mayer and Miiller, 2nd ed., 1885).

F. 23

354 GENERAL RESOLUTION OF SURFACE [180.

the upper sheet ; no one of them, except at its initial and its final points,
meets any of the cuts already made. This system of p - 1 cuts we denote
by c% , GS, . . . , Cp .

Because br^ is a cross-cut, the cross-cut cr changes ar (hitherto a loop-
cut) into a cross-cut when cr and ar are combined into a single cut.

It is evident that no one of these cuts divides the surface into distinct
pieces; and thus we have a system of 2p cross-cuts resolving the two-sheeted
surface of connectivity 2p+I into a surface that is simply connected. The
cross-cuts in order* are

Oj, &j, C2 and aa, 62, c3 and as, bs, ...,cp and ap, bp.

181. This resolution of a general two-sheeted surface suggests f Rie-

mann's general resolution of a surface with any (finite) number of sheets.

As before, we assume that the surface is closed and has a single boundary

and that its class is p, so that 2p cross-cuts are necessary for its resolution

into one that is simply connected.

Make a cut in the surface such as not to divide it into distinct pieces;
and let it begin and end in the boundary. It is a cross-cut, say ^ ; it
changes the number of boundary-lines to 2 and it reduces the connectivity
of the cut surface to 2p.

Since the surface is connected, we can pass in the surface along a
continuous line from one edge of the cut ^ to the opposite edge. Along
this line make a cut 6j : it is a cross-cut, because it begins and ends in
the boundary. It passes from one edge of c^ to the other, that is, from one
boundary-line to another. Hence, as in Prop. II. of § 164, it does not divide '
the surface into distinct pieces; it changes the number of boundaries to 1
and it reduces the connectivity to 1p — 1.

The problem is now the same as at first, except that now only
2« — 2 cross-cuts are necessary for the required resolution. We make a
loop-cut a.2, not resolving the surface into distinct pieces, and a cross-cut
d from a point of a2 to a point on the boundary at 6j ; then Cj and a,2> taken
together, constitute a cross-cut that does not resolve the surface into distinct
pieces. It therefore reduces the connectivity to 2p — 2 and leaves two pieces
of boundary.

The surface being connected, we can pass in the surface along a continuous
line from one edge of a» to the opposite edge. Along this line we make a cut
b.2, evidently a cross-cut, passing, like h in the earlier case, from one
boundary-line to the other. Hence it does not divide the surface into

* See Neumann, pp. 178 — 182; Prym, Zur Thcorie der Fwwtionen in einer zweiblattrigen
Flfahe, (1866).

+ Riemann, Ges. Werke, pp. 122, 123 ; Neumann, pp. 182—185.

181.] BY CROSS-CUTS 355

distinct pieces; it changes the number of boundaries to 1 and it reduces
the connectivity to 2p — 3.

Proceeding in p stages, each of two cross-cuts, we ultimately obtain a
simply connected surface with a single boundary ; and the general effect on
the original unresolved surface is to have a system of cross-cuts somewhat of
the form

Fig. 63.

The foregoing resolution is called the canonical resolution of a Riemann's
surface.

Ex. 1. Construct the Riemann's surface for the equation

w3 + z3 — 3awz— 1,

both for a = 0 and for a different from zero; and resolve it by cross-cuts into a simply
connected surface with a single boundary, shewing a complete system of irreducible simple
circuits on the unresolved surface.

Ex. 2. Shew that the Riemann's surface for the equation

_
(z-c)(z-d)

is of class p = 2- indicate the possible systems of branch-lines, and, for each system,
resolve the surface by cross-cuts into a simply connected surface with a single boundary.

(Burnside.)

182. Among algebraical equations with their associated Riemann's
surfaces, two general cases of great importance and comparative simplicity
distinguish themselves.. The first is that in which the surface is two-
sheeted ; round each branch-point the two branches interchange. The
second is that in which, while the surface has a finite number of sheets
greater than two, all the branch-points are of the first order, that is, are
such that round each of them only two branches of the function interchange.
The former has already been considered, in so for as concerns the surface ;
we now proceed to the consideration of the latter.

The equation is f(w, z) = 0,

of degree n in w; and, for our present purpose, it is convenient to regard

0 as an equation corresponding to a generalised plane curve of degree n
so that no term in / is of dimensions higher than n.

The total number of branch-points has been proved, in § 98, to be

w(w-l)-28-2«,

23—2

356 DEFICIENCY [182.

where S is the number of points which are the generalisation of double
points on the curve with non-coincident tangents and K is the number
of double points on the curve with coincident tangents. Round each of
these branch-points, two branches of w interchange and only two, so that
all the numbers mq of § 178 are equal to 2 ; hence the ramification
H is

2 [n (n - 1) - 2S - 2/e} - [n (n - 1) - 2S - 2*},

that is, n=w(n-l)-28-2«.

The connectivity of the surface is therefore

w (n - 1) - 28 - 2* - 2n + 3 ;
and therefore the class p of the surface is

£(n-l)(»-2)-8-«.

Now this integer is known* as the deficiency of the curve; and therefore it
appears that the deficiency of the curve is the same as the class of the Riemann
surface associated with its equation, and also is the same as the class of its
equation.

Moreover, the number of branch-points of the original equation is fl, that

is,

n - 2

Note. The equality of these numbers, representing the deficiency and
the class, is one among many reasons that lead to the close association of
algebraic functions (and of functions dependent on them) with the theory of
plane algebraic curves, in the investigations of Nb'ther, Brill, Clebsch and
others, referred to in §§ 191, 242.

183. With a view to the construction of a canonical form of Riemann's
surface of class p for the equation under consideration, it is necessary to
consider in some detail the relations between the branches of the functions
as they are affected by the branch- points.

The effect produced on any value of the function by the description of a
small circuit, enclosing one branch-point (and only one), is known. But
when the small circuit is part of a loop, the effect on the value of the
function with which the loop begins to be described depends upon the form
of the loop; and various results (e.g. Ex. 1, § 104) are obtained by taking
different loops. In the first form (§ 175) in which the branch-lines were
established as junctions between sheets, what was done was the equivalent

* Salmon's Higher Plane Curves, §§ 44, 83; Clebsch's Vorlesungen iiber Geometrie, (edited
by Lindemann), t. i, pp. 351 — 429, the German word used instead of deficiency being Geschlecht.
The name 'deficiency' was introduced by Cayley in 1865: see Proc. Land. Math. Soc., vol. i.,
" On the transformation of plane curves."

183.] LOOPS 357

of drawing a number of straight loops, which had one extremity common to
all and the other free, and of assigning the law of junction according to the
law of interchange determined by the description of the loop. As, however,
there is no necessary limitation to the forms of branch-lines, we may draw
them in other forms, always, of course, having branch-points at their free
extremities ; and according to the variation in the form of the branch-line,
(that is, according to the variation in the form of the corresponding loop
or, in other words, according to the deformation of the loop over other
branch-points from some form of reference), there will be variation in the law
of junction along the branch-lines.

There is thus a large amount of arbitrary character in the forms of the
branch-lines, and consequently in the laws of junction along the branch-lines,
of the sheets of a Riemann's surface. Moreover, the assignment of the n
branches of the function to the n sheets is arbitrary. Hence a consider
able amount of arbitrary variation in the configuration of a Riemann's
surface is possible within the limits imposed by the invariance of its
connectivity. The canonical form will be established by making these
arbitrary elements definite.

184. After the preceding explanation and always under the hypothesis
that the branch-points are simple, we shall revert temporarily to the use of
loops and shall ultimately combine them into branch-lines.

When, with an ordinary point as origin, we construct a loop round a
branch-point, two and only two of the values of the function are affected
by that particular loop ; they are interchanged by it ; but a different form of
loop, from the same origin round the same branch-point, might affect some
other pair of values of the function.

To indicate the law of interchange, a symbol will be convenient. If the
two values interchanged by a given loop be Wi and wm, the loop will be
denoted by im ; and i and ra will be called the numbers of the symbol of that
loop.

For the initial configuration of the loops, we shall (as in § 175) take an
ordinary point 0 : we shall make loops beginning at 0, forming them in the
sequence of angular succession of the branch-points round 0 and drawing the
double linear part of the loop as direct as possible from 0 to its branch-point :
and, in this configuration, we shall take the law of interchange by a loop to
be the law of interchange by the branch-point in the loop.

In any other configuration, the symbol of a loop round any branch-point
depends upon its form, that is, depends upon the deformation over other
branch-points which the loop has suffered in passing from its initial form.
The effect of such deformation must first be obtained : it is determined by
the following lemma : —

358

MODIFICATION

[184.

When one loop is deformed over another, the symbol of the deformed loop is
unaltered, if neither of its numbers or if both of its numbers occur in the
symbol of the unmoved loop ; but if, before deformation, the symbols have one
number common, the new symbol of the deformed loop is obtained from the old
symbol by substituting, for the common number, the other number in the symbol
of the unmoved loop.

The sufficient test, to which all such changes must be subject, is that
the effect on the values of the function at any point of a contour enclosing
both branch-points is the same at that point for all deformations into two
loops, Moreover, a complete circuit of all the loops is the same as a contour
enclosing all the branch-points; it therefore (Cor. III. § 90) restores the initial
value with which the circuit began to be described.

Obviously there are three cases.

First, when the symbols have no number common : let them be mn, rs.
The branch-point in the loop rs does not affect wm or wn: it is thus effectively
not a branch-point for either of the values wm and wn; and therefore (§ 91)
the loop mn can be deformed across the point, that is, it can be deformed
across the loop mn.

Secondly, when the symbols are the same : the symbol of the deformed
loop must be unaltered, in order that the contour embracing only the two
branch-points may, as it should, restore after its complete description each of
the values affected.

Thirdly, when the symbols have one number common : let 0 be any
point and let the loops be OA, OB in any given position such as (i), Fig. 64,
with symbols mr, nr respectively. Then OB may be deformed over OA as
in (ii), or OA over OB as in (iii).

Fig. 64

The effect at 0 of a closed circuit, including the points A and B and
described positively beginning at 0, is, in (i) which is the initial configura
tion, to change wm into wr, wr into wn, wn into wm\ this effect on the
values at 0, unaltered, must govern the deformation of the loops.

The two alternative deformations (ii) and (iii) will be considered separately.

When, as in (ii), OB is deformed over OA, then OA is unmoved and
therefore unaltered : it is still mr. Now, beginning at 0 with wm, the loop

184.] OF LOOPS 359

OA changes wm into wr: the whole circuit changes wm into wr, so that OB
must now leave wr unaltered. Again, beginning with wn, it is unaltered by
0 A, and the whole circuit changes wn into wm : hence OB must change wn
into wm, that is, the symbol of OB must be inn. And, this being so, an
initial wr at 0 is changed by the whole circuit into wn, as it should be.
Hence the new symbol mn of the deformed loop OB in (ii) is obtained from
the old symbol by substituting, for the common number r, the other number
in in the symbol of the unmoved loop OA.

We may proceed similarly for the deformation in (iii) ; or the new symbol
may be obtained as follows. The loop 0 A in (iii) may be deformed to the
form in (iv) without crossing any branch- point and therefore without
changing its symbol. When this form of the loop is described in the
positive direction, wn initially at 0 is changed into w.r after the first loop
OB, for this loop has the position of OB in (i), then it is changed into wm
after the loop OA, for this loop has the position of OA in (i), and then wm is
unchanged after the second (and inner) loop OB. Thus wn is changed into
wm, so that the symbol is mn, a symbol which is easily proved to give the
proper results with an initial value wm or wr for the whole contour. This
change is as stated in the theorem, which is therefore proved.

Ex. If the deformation from (i) to (ii) be called superior, and that from (i) to (iii)
inferior, then x successive superior deformations give the same loop-configuration, in
symbols and relative order for positive description, as 6 — &• successive inferior deform
ations.

COROLLARY. A loop can be passed unchanged over two lo.ops that have the
same symbol.

Let the common symbol of the unmoved loops be mn. If neither number
of the deformed loop be m or n, passage over each of the loops mn makes no
difference, after the lemma ; likewise, if its symbol be mn. If only one of its
numbers, say n, be in mn, its symbol is nr, where r is different from m. When
the loop nr is deformed over the first loop mn, its new symbol is mr ; when
this loop mr is deformed over the second loop mn, its new symbol is nr, that
is, the final symbol is the same as the initial symbol, or the loop is unchanged.

185. The initial configuration of the loops is used by Clebsch and
Gordan to establish their simple cycles and thence to deduce the periodi
city of the Abelian integrals connected with the equation f(w, z) = 0,
without reference to the Riemann's surface ; and this method of treating
the functions that arise through the equation, always supposed to have
merely simple branch-points, has been used by Casorati* and Liiroth-J-.

We can pass from any value of w at the initial point 0 to any other

* Annali di Matematica, 2da Ser., t. iii, (1870), pp. 1 — 27.
t Abh. d. K. bay. Akad. t. xvi, i Abth., (1887), pp. 199—241.

360 CYCLES OF LOOPS [185.

value by a suitable series of loops ; because, were it possible to inter
change the values of only some of the branches, an equation could be
constructed which had those branches for its roots. The fundamental
equation could then be resolved into this equation and an equation having
the rest of the branches for its roots : that is, the fundamental equation
would cease to be irreducible.

We begin then with any loop, say one connecting wl with w2. There
will be a loop, connecting the value w3 with either wl or w.,; there will
be a loop, connecting the value wt with either w1} w.2, or w3; and so on,
until we select a loop, connecting the last value wn with one of the other
values. Such a set of loops, n — 1 in number, is called fundamental.

A passage round the set will not at the end restore the branch with
which the description began. When we begin with any value, any other
value can be obtained after the description of properly chosen loops of the
set.

Any other loop, when combined with a set of fundamental loops, gives
a system the description of suitably chosen loops of which restores some
initial value ; only two values can be restored by the description of loops
of the combined system. Thus if the loops in order be 12, 13, 14,..., In
and a loop qr be combined with them, the value wq is changed into Wj_ by
Iq, into wr by Ir, into wq by qr; and similarly for wr. Such a combination
of n loops is called a simple cycle.

The total number of branch-points, a.nd therefore of loops, is (§ 182)

2 {/> + (»-!)};

and therefore the total number of simple cycles is 2p+n — l. But these
simple cycles are not independent of one another.

In the description of any cycle, the loops vary in their operation
according to the initial value of w : and, for two different initial values of
w, no loop is operative in the same way. For otherwise all the preceding
and all the succeeding loops would operate in the same way and would
lead, on reversal, to the same initial value of w. Hence a loop of a given
cycle can be operative in only two descriptions, once when it changes, say, wi
into Wj, and the other when it changes Wj into W{.

Now consider the circuit made up of all the loops. When wl is taken as
the initial value, it is restored at the end : and in the description only a
certain number of loops have been operative : the cycle made up of these loops
can be resolved into the operative parts of simple cycles, that is, into simple
cycles : hence one relation among the simple cycles is given by the considera
tion of the operative loops when the whole system of the loops is described
with an initial value.

Similarly when any other initial value is taken ; so that apparently there

185.] LUROTH'S THEOREM 361

are n relations, one arising from each initial value. These n relations are not
independent : for a simultaneous combination of the operations of all the
loops in all the circuits leads to an identically null effect (but no smaller
combination would be effective), for each loop is operative twice (and only
twice) with opposite effects, shewing that one and only one of the relations is
derivable from the remainder. Hence there are n — 1 independent relations
and therefore* the number of independent simple cycles is 2p.

186. We now proceed to obtain a typical form of the Riemann's surface
by deforming the initial configuration of the loops into a typical configu
ration f. The final arrangement of the loops is indicated by the two
theorems : —

I. The loops can be made in pairs in which all loop-symbols are of the
form (m, in + I), for m = 1, 2, ... , n — 1. (With this configuration, w1 can be
changed by a loop only into w.2, w., by a loop only into w3, and so on in
succession, each change being effected by an even number of loops.) This
theorem is due to Liiroth.

II. The loops can be made so that there is only one pair 12, only one
pair 23, . . . , only one pair ()i — 2, n — 1 ), and the remaining p + I pairs are
(n — 1, n). This theorem is due to Clebsch.

187. We proceed to prove Liiroth's theorem, assuming that the loops
have the initial configuration of § 184.

Take any loop 12, say OA : beginning it with w1} describe loops positively
and in succession ; then as the value wl is restored sooner or later, for it
must be restored by the circuit of all the loops, let it be restored first by a
loop OB, the symbol of OB necessarily containing the number 1. Between OA
and OB there may be loops whose symbols contain 1 but which have been
inoperative. Let each of these in turn be deformed so as to pass back over
all the loops between its initial position and OA ; and then finally over OA.
Before passing over OA its symbol must contain 1, for there is no loop over
which it has passed that, having 1 in its symbol, could make it drop 1 in the
passage ; but it cannot contain 2, for, if it did, the effect of OA and the
deformed loop would be to restore 1, an effect that would have been
caused in the original position, contrary to the hypothesis that OB is the
first loop that restores 1. Hence after it has passed over OA its symbol
no longer contains 1.

* Clebsch und Gordan, Theorie der AbcVschen Functional, p. 85.
t The investigation is based upon the following memoirs : —

Liiroth, "Note liber Verzweigungsschnitte und Querschnitte in einer Riemann'scheu Fla'che,"
Math. Ann., t. iv, (1871), pp. 181—184; "Ueber die kanonischen Perioden der Abel'schen
Integrate," Abh. d. K. bay. Akad., t. xv, ii Abth., (1885), pp. 329—366.

Clebsch, "Zur Theorie der Riemann'schen Flachen," Math. Ann., t. vi, (1873), pp. 216—230.
Clifford, " On the canonical form and dissection of a Riemann's Surface," Loud. Math. Soc.
Proc., vol. viii, (1877), pp. 292—304.

362 LUROTH'S THEOREM [187.

Next, pass OB over the loops between its initial position and OA but not
over OA : its symbol must be 12 in the deformed position since w, is restored
by the loop OB. Then OA and the deformed loop OB are each 12 ; hence each
of the loops, between the new position and the old position of OB, can be passed
over OA and the new loop OB without any change in its symbol. There are
therefore, behind OA, a series of loops that do not affect w^ Thus the loops

are

(a) loops behind OA not affecting wlf (b) OA, OB each 12,
(c) other loops beyond the initial position of OB.

Begin now with wa at the loop OB and again describe loops positively
and in succession: then w.2 must be restored sooner or later. It may be
only after OA is described, so that there has been a complete circuit of
all the loops ; or it may first be by an intermediate loop, say 00.

For the former case, when OA is the first loop by which w.2 is restored,
we deform as follows. Deform all loops affecting w1} which lie between
OB and OA, in the positive direction from OB back over other loops and
over OB. The symbol of each just before its deformation contains 1 but
not 2, and therefore after its deformation it does not contain 1. Moreover
just after OB is described, wl is the value, and just before OA is described,
wl is the value ; hence the intermediate loops, which have affected wlf
must be even in number. Let OG be the first after OB which affects wlt
and let the symbol of OG be Ir. Then beginning OG with w1} the value
Wj_ must be restored by a complete circuit of all the loops, that is, it
must be restored by OB] and therefore the value must be Wi when
beginning OA, or Wj. must be restored before OA. Let OH be the first
loop after OG to restore w^, then, by proceeding as above, we can deform
all the loops between OG and OH over OG, with the result that no such
deformed loop affects w± and that OG and OH are both Ir. Hence all
the loops affecting w1 can be arranged in pairs having the same symbol

Since OG and OH are a pair with the same symbol, every loop between
OB and OG can be passed unchanged over OG and OH together. When
this is done, pass OG over OB so that it becomes 2r, and then OH over
OB so that it also is 2r. Thus these deformed loops OG, OH are a pair
2r; and therefore OA can, without change, be deformed over both so as
to be next to OB. Let this be done with all the pairs ; then, finally, we
have

(a) loops not affecting wlt (b) a pair with the symbol 12,

(c) pairs affecting w, and not w1} (d) loops not affecting w±.

We thus have a pair 12 and loops not affecting tv^ so that such a change
has been effected as to make all the loops affecting w1 possess the symbol 12.
For the second case, when OC is the first loop to restore w,, the

187.] ON CONFIGURATION OF LOOPS 363

value with which the loop OB whose symbol is 12 began to be described, we
treat the loops between OB and 00 in a manner similar to that adopted in
the former case for loops between OA and OB ; so that, remembering that
now w.2 instead of the former wl is the value dealt with in the recurrence, we
can deform these loops into

(a) loops behind OB which change wl but not w.2,

(b) OB and OC, the symbol of each of which is 12.

Now OB was next to OA ; hence the set (a) are now next to OA. Each of
them when passed over OA drops the number 1 from its symbol and so the
whole system now consists of

(a) loops behind OA not affecting wlt (b) OA, OB, 00 each of which
is 12, (c) other loops.

Begin again with the value wl before OA. Before OC the value is w^\
and the whole circuit of the loops must restore w1} which must therefore
occur before OA. Let OD be the first loop by which w^ is restored. Then
treating the loops between OC and OD, as formerly those between the initial
positions of OA and OB were treated, we shall have

(a) loops behind OA not affecting wn (b) OA, OB each being 12,

(c) loops between OB and OC not affecting w1} (d) OC, OD each

being 12, (e) other loops.

Except that fewer loops affecting wl have to be reckoned with, the con
figuration is now in the same condition as at the end of the first stage.
Proceeding therefore as before, we can arrange that all the loops affecting wt
occur in pairs with the symbol 12. Moreover, each of the loops in the set
(c) can be passed unchanged over OA and OB ; so that, finally, we have

(a) pairs of loops with the symbol 12, (&') loops not affecting w^.

We keep (a) in pairs, so that any desired deformation of loops in (&') over
them can be made without causing any change; and we treat the set (6') in
the same manner as before, with the result that the set (b') is replaced by

(6) pairs of loops with the symbol 23, (c') loops not affecting WL or w.2.

And so on, with the ultimate result that Hie loops can be made in pairs in
which each symbol is of the form (in, m + 1) for m = l, ... , n — 1.

188. We now come to Clebsch's Theorem that the loops thus made can
be so deformed that there is only one pair 12, only one pair 23, and so on,
until the last symbol (n— 1, n), which is the common symbol of p+ 1 pairs.

This can be easily proved after the establishment of the lemma that, if
there be two pairs 1 2 and one pair 23, the loops can be deformed into one pair
12 and two pairs 23.

364 CLEBSCH'S THEOREM [188.

The actual deformation leading to the lemma is shewn in the accompany
ing scheme : the deformations implied by the

i r i c xi i fj. 12 12 12 12 26 2o

continuous lines are those ot a loop from the left

to the right of the respective lines, and those 12 12 12 23 13 23

implied by the dotted lines are those of a loop 12 12 £3 13 13 23
from the right to the left of the respective lines.

., ,-, . L2 L2 id lo ^o AO

It is interesting to draw figures, representing

the loops in the various configurations. 12 23 12 13 23 23

By the continued use of this lemma we can 12 23 23 12 23 23

change all but one of the pairs 12 into pairs 23, 12 12 23 23 23 23
all but one of the pairs 23 into pairs 34, and

so on, the final configuration being that there are one pair 12, one pair 23, ...
and p + 1 pairs (n - 1, n). Thus Clebsch's theorem is proved.

189. We now proceed to the construction of the Biemann's surface.

Each loop is associated with a branch-point, and the order of interchange
for passage round the branch-point, by means of the loop, is given by the
numbers in the symbol of the loop.

Hence, in the configuration which has been obtained, there are two branch
points 12: we therefore connect them (as in § 176) by a line, not necessarily
along the direction of the two loops 12 but necessarily such that it can,
without passing over any branch-point, be deformed into the lines of the
two loops; and we make this the branch-line between the first and the
second sheets. There are two branch-points 23 : we connect them by a line
not meeting the former branch-line, and we make it the branch-line between
the second and the third sheets. And so on, until we come to the last two
sheets. There are *2p + 2 branch-points n-l,n: we connect these in pairs
(as in § 176) by p + 1 lines, not meeting one another or any of the former
lines, and we make them the p + 1 branch-lines between the last two sheets.

It thus appears that, when the winding -points of a Riemann's surface with
n sheets of connectivity 2p + 1 are all simple, the surface can be taken in such
a form that there is a single branch-line between consecutive sheets except for tlie
last two sheets : and between the last two sheets there are p+l branch-lines.
This form of Riemann's surface may be regarded as the canonical form for a
surface, all the branch-points of which are simple.

Further, let AB be a branch-line such as 12. Let two points P and Q
be taken in the first sheet on opposite sides of AB, so that PQ in space is
infinitesimal ; and let P' be the point in the second sheet determined by the
same value of z as P, so that P'Q in the sheet is infinitesimal. Then the
value Wi at P is changed by a loop round A (or round B) into a value at Q
differing only infinitesimally from w.2, which is the value at P' : that is, the
change in the function from Q to P' is infinitesimal. Hence the value of the
function is continuous across a line of passage from one sheet to another.

190.] CANONICAL SURFACE 365

190. The class of the foregoing surface is p ; and it was remarked, in
§ 170, that a convenient surface of reference of the same class is that of a
solid sphere with p holes bored through it. It is, therefore, proper to in
dicate the geometrical deformation of a Riemann's surface of this canonical
form into a p-ho\ed sphere.

The Riemann's surface consists of n sheets connected chainwise each with
a single branch-line to the sheet on either side of it, except that the first is
connected only with the second and that the last two have p + 1 branch-
lines. We may also consider the whole surface as spherical and the sequence
of the sheets from the inside outwards : and the outmost sheet can be con
sidered as bounded.

Let the branch-line between the first and the second sheets be made to
lie along part of a great circle. Let the first sheet of the Riemann's surface
be reflected in the plane of this great circle : the line becomes a long
narrow hole along the great circle, and the reflected sheet becomes a large
indentation in the second sheet. Reversing the process of § 169, we can
change the new form of the second sheet, so that it is spherical again : it is
now the inmost of the n — 1 sheets of the surface, the connectivity and the
ramification of which are unaltered by the operation.

Let this process be applied to each surviving inner sheet in succession.
Then, after n-2 operations, there will be left a two-sheeted surface ; the
outer sheet is bounded and the two sheets are joined by p + 1 branch-
lines ; so that the connectivity is still 2|) + 1. Let these branch-lines be
made to lie along a great circle: and let the inner surface be reflected
in the plane of this circle. Then, after the reflexion, each of the branch-lines
becomes a long narrow hole along the great circle ; and there are two
spherical surfaces which pass continuously into one another at these holes,
the outer of the surfaces being bounded. By stretching one of the holes
and flattening the two surfaces, the new form is that of a bifacial flat
surface: each of the p holes then becomes a hole through the body
bounded by that surface ; the stretched hole gives the extreme geo
metrical limits of the extension of the surface, and the original boundary of
the outer surface becomes a boundary hole existing in only one face. The
body can now be distended until it takes the form of a sphere, and the final
form is that of the surface of a solid sphere with p holes bored through it
and having a single boundary.

This is the normal surface of reference (§ 170) of connectivity 2p + 1.

As a last ground of comparison between the Riemann's surface in its
canonical form and the surface of the bored sphere, we may consider the
system of cross-cuts necessary to transform each of them into a simply
connected surface.

We begin with the spherical surface. The simplest irreducible circuits

366 DEFORMATION [190.

are of two classes, (i) those which go round a hole, (ii) those which go through
a hole; the cross-cuts, 2p in number, which make the surface simply con
nected, must be such as to prevent these irreducible circuits.

Round each of the holes we make a cut a, the first of them beginning
and ending in the boundary : these cuts prevent circuits through the holes.
Through each hole we make a cut 6, beginning and ending at a point in the
corresponding cut a : we then make from the first b a cut Ci to the second a,
from the second 6 a cut c2 to the third a, and so on. The surface is then
simply connected : a± is a cross-cut, 6j is a cross-cut, Cj + a2 is a cross-cut,
&2 is a cross-cut, c2 + a3 is a cross-cut, and so on. The total number is
evidently 2p, the number proper for the reduction ; and it is easy to verify
that there is a single boundary.

To compare this dissection with the resolution of a Riemann's surface by
cross-cuts, say of a two-sheeted surface (the rc-sheeted surface was trans
formed into a two-sheeted surface), it must be borne in mind that only p of
the p + 1 branch-lines were changed into holes and the remaining one, which,
after the partial deformation, was a hole of the Riemann's surface, was
stretched out so as to give the boundary.

It thus appears that the direction of a cut a round a hole in the normal
surface of reference is a cut round a branch-line in one sheet, that is, it is a
cut a as in the resolution (§ 180) of the Riemann's surface into one that is
simply connected.

Again, a cut b is a cut from a point in the boundary across a cut a and
through the hole back to the initial point ; hence, in the Riemann's surface,
it is a cut from some one assigned branch-line across a cut ar, meeting the
branch-line surrounded by ar, passing into the second sheet and, without
meeting any other cut or branch-line in that surface, returning to the initial
point on the assigned branch-line. It is a cut b as in the resolution of the
Riemann's surface.

Lastly, a cut c is made from a cut b to a cut a. It is the same as in the
resolution of the Riemann's surface, and the purpose of each of these cuts is
to change each of the loop-cuts a (after the first) into cross-cuts.

A simple illustration arises in the case of a two-sheeted Riemann's surface, of class£> = 2.
The various forms are :

(i) the surface of a two-holed sphere, with the directions of cross-cuts that resolve it
into a simply connected surface; as in (i), Fig. 65, B, K being at opposite edges of
the cut Cj where it meets a.2: II, C at opposite edges where it meets b^. and so on;

(ii) the spherical surface, resolved into a simply connected surface, bent, stretched,
and flattened out ; as in (ii), Fig. 65;

(iii) the plane Riemann's surface, resolved by the cross-cuts ; as in Fig. 63, p. 355.

Numerous illustrations of transformations of Riemann's surfaces are given by
Hofmann, Methodik der stetigen Deformation von zweibldttrigan Riemann'schen Fliichen,
(Halle a. S., Nebert, 1888).

190.]

OF RIEMANNS SURFACES

367

191. We have seen that a bifacial surface with a single boundary can be
deformed, at least geometrically, into any other bifacial surface with a single
boundary, provided the two surfaces have the same connectivity ; and the
result is otherwise independent of the constitution of the surface, in regard
to sheets and to form or position of branch-lines. Further, in all the geo
metrical deformations adopted, the characteristic property is the uniform
correspondence of points on the surfaces.

Now with every Riemann's surface, in its initial form, an algebraical
equation /(w, z) = Q is associated; but when deformations of the surface
are made, the relations that establish uniform correspondence between
different forms, practically by means of conformal representation, are often
of a transcendental character (Chap. XX.). Hence, when two surfaces are
thus equivalent to one another, and when points on the surfaces are
determined solely by the variables in the respective algebraical equations,
no relations other than algebraical being taken into consideration, the
uniform correspondence of points can only be secured by assigning a new
condition that there be uniform transformation between the variables w and
2 of one surface and the variables w' and z' of the other surface. And, when
this condition is satisfied, the equations are such that the deficiencies of the
two (generalised) curves represented by the equations are the same, because
they are equal to the common connectivity. It may therefore be expected
that, when the variables in an equation are subjected to uniform transfor
mation, the class of the equation is unaltered ; or in other words that the
deficiency of a curve is an invariant for uniform transformation.

This inference is correct : the actual proof is more directly connected
with geometry and the theory of Abelian functions, and must be sought
elsewhere*. The result is of importance in justifying the adoption of a
simple normal surface of the same class as a surface of reference.

* Clebsch's Vorlcsungen iiber Geometric, t. i, p. 459, where other references are given; Salmon's
Higher Plane Curves, pp. 93, 319; Clebsch und Gordan, Theorie der Abel'schen Functionen,
Section 3; Brill, Math. Ann., t. vi, pp. 33 — 65.

CHAPTER XVI.

ALGEBRAIC FUNCTIONS AND THEIR INTEGRALS.

192. IN the preceding chapter sufficient indications have been given as
to the character of the Riemann's surface on which the ?i-branched function
w, determined by the equation

/(»,*)=(>,

can be represented as a uniform function of the position of the variable. It
is unnecessary to consider algebraically multiform functions of position on
the surface, for such multiformity would merely lead to another surface of
the same kind, on which the algebraically multiform functions would be
uniform functions of position ; transcendental ly multiform functions of
position will arise later, through the integrals of algebraic functions. It
therefore remains, at the present stage, only to consider the most general
uniform function of position on the Riemann's Surface.

On the other hand, it is evident that a Riemann's Surface of any number
of sheets can be constructed, with arbitrary branch-points and assigned
sequence of junction ; the elements of the surface being subject merely to
general laws, which give a necessary relation between the number of sheets,
the ramification and the connectivity, and which require the restoration of
any value of the function after the description of some properly chosen
irreducible circuit. The essential elements of the arbitrary surface, and the
merely general laws indicated, are independent of any previous knowledge
of an algebraical equation associated with the surface ; and a question arises
whether, when a Riemann's surface is given, an associated algebraical equa
tion necessarily exists.

Two distinct subjects of investigation, therefore, arise. The first is the
most general uniform function of position on a surface associated with a given
algebraical equation, and its integral ; the second is the discussion of the
existence of functions of position on a surface that is given independently

192.] FUNCTIONS OF POSITION 369

of an algebraical equation. Both of them lead, as a matter of fact, to the
theory of transcendental (that is, non-algebraical) functions of the most
general type, commonly called Abelian transcendents. But the first is,
naturally, the more direct, in that the algebraical equation is initially given :
whereas, in the second, the prime necessity is the establishment of the so-
called Existence-Theorem — that such functions, algebraical and transcen
dental, exist.

193. Taking the subjects of investigation in the assigned order, we
suppose the fundamental equation to be irreducible, and algebraical as
regards both the dependent and the independent variable ; the general form
is therefore

WnG0 (Z) + W"-^ (Z) + . . . + wGn^ (Z) + Gn (z) = 0,

the coefficients G0(z), G^(z\ ..., Gn(z) being rational, integral, algebraical
functions.

The infinities of w are, by § 95, the zeros of G0 (z) and, possibly, z= oo .
But, for our present purpose, no special interest attaches to the infinity of a
function, as such ; we therefore take wG0 (z) as a new dependent variable,
and the equation then is

/ (w, z) = wn + wn~l g1(z)+...+ wgn^ (z) + gn (z) = 0,

in which the functions g (z) are rational, integral, algebraical functions
of z.

The distribution of the branches for a value of z which is an ordinary
point, and the determination of the branch-points together with the cyclical
grouping of the branches round a branch-point, may be supposed known.
When the corresponding w-sheeted Riemann's surface (say of connectivity
2p + 1) is constructed, then w; is a uniform function of position on the
surface.

Now not merely w, but every rational function of w and z, is a uniform
function of position on the surface; and its branch-points (though not
necessarily its infinities) are the same as that of the function w.

Conversely, every uniform function of position on the Riemanris surface,
having accidental singularities and infinities only of finite order, is an
algebraical rational function of w and z. The proof* of this proposition,
to which we now proceed, leads to the canonical expression for the most
general uniform function of position on the surface, an expression which is
used in Abel's Theorem in transcendental integrals.

Let w' denote the general uniform function, and let w/, w/, ..., wn' denote
the branches of this function for the points on the n sheets determined by

* The proof adopted follows Prym, Crelle, t. Ixxxiii, (1877), pp. 251—261 ; see also Klein,
Ueber Riemann's Theorie der algebraischen Functionen und ihrer Integrate, p. 57.

F. 24

370

UNIFORM FUNCTIONS OF POSITION

[193.

the algebraical magnitude z\ and let wlt w.2, ...,wn be the corresponding
branches of w for the magnitude z. Then the quantity

WfWi + W*Wz + • • • + Wn*Wn,

where s is any positive integer, is a symmetric function of the possible values
of wsw' ; it has the same value in whatever sheet z may lie and by whatever
path z may have attained its position in that sheet ; the said quantity is there
fore a uniform function of z. Moreover, all its singularities are accidental in
character, by the initial hypothesis as to w' and the known properties of
w ; they are finite in number ; and therefore the uniform function of z is
algebraical. Let it be denoted by h, (z), which is an integral function only
when the singularities are for infinite values of z ; then

wfWi + wfWz + . . . + wn*wn' = hs (z),

an equation which is valid for any positive integer s, there being of course
the suitable changes among the rational integral algebraical functions h (z} for
changes in s. It is unnecessary to take s ^ n, when the equations for the
values 0,1, ..., n — 1 of s are retained: for the equations corresponding to
values of s ^ n can be derived, from the n equations that are retained, by
using the fundamental equation determining w.

Solving the equations

-iWi + W2W2' -I- . . . + WnWn' = h-i

Wf-hVi + ... + Wn"~W = hn-i (z\

to determine w/, we have

1,

1, .-, 1

=

h0(z), 1, .

.., 1

w,,

Wo, ..., Wn

/? ^.2^ Wo

.., w»

w 2

w.?, ..., wn2

*,(*), W22' •

w 2

wn-iy

w,/*-1, , ., Wn"-1

^ (^) Won~1

w n—1

The right-hand side is evidently divisible by the product of the differences
of w2, w3, ..., wn; and this product is a factor of the coefficient of w/.
Then, if

n

(w — w2) (w — w3) ... (w — wn) = S krwn~r,

r=l

where ki is unity, we have, on removing the common factor,

1 (Wx - W2) (W1 — W3) . . . (Wj — Wn)

193.] ON A RIEMANN'S SURFACE 371

But / (w, z) = (w — w^ (w — w2)...(w — wn),

so that k2 = wl + gl(z),

k3 = wf + W& (z) + g2 (z),

kn = wf-1 + w^-g, (z) + . . . + gn^ (z).

When these expressions for k are substituted in the numerator of the ex
pression for Wi, it takes the form of a rational integral algebraical function
of w of degree n — 1 and of z, say

h0 (z) w^-1 -f H, (z) <l~2 + . . . + Hn_, (z) w, + Hn^ (z).

The denominator is evidently df/dwl, when w is replaced by wl after differen
tiation, so that we now have

df/dw,

The corresponding form holds for each of the branches of w': and therefore we
have

) wn~* + . . . + ffn-i (z)

df/dw

=

nwn~l + (n - 1) wn~zgl (z)+...+ gn_, (z) '

so that w' is a rational, algebraical, function of w and z. The proposition is
therefore proved.

By eliminating w between f (w, z) = 0 and the equation which expresses
w' in terms of w and z, or by the use of § 99, it follows that w satisfies an
algebraical equation

F(w',z) = 0,

where F is of order n in w' ; the equations / (w, z) — 0 and F (w', z) = 0 have
the same Riemann's surface associated with them*.

194. It thus appears that there are uniform functions of position on
the Riemann's surface just as there are uniform functions of position in
a plane. The preceding proposition is limited to the case in which the
infinities, whether at branch-points or not, are merely accidental ; had the
function possessed essential singularities, the general argument would still
be valid, but the forms of the uniform functions h (z) would no longer be
algebraical. In fact, taking account of the difference in the form of the
surface on which the independent variable is represented, we can extend
to multiform functions, which are uniform on a Riemann's surface, those
propositions for uniform functions which relate to expansion near an ordinary
point or a singularity or, by using the substitution of § 93, a branch
singularity, those which relate to continuation of functions, and so on ;

* See § 191. Functions related to one another, as w and w' are, are called gleichverzweigt,
Riemann, p. 93.

24—2

372 ALGEBRAICAL FUNCTIONS [194.

and their validity is not limited, as in Cor. VI., § 90, to a portion of the
surface in which there are no branch-points.

Thus we have the theorem that a uniform algebraical function of position
on the Riemanns surface has as many zeros as it has infinities.

This theorem may be proved as follows.

The function is a rational algebraical function of w and z. If it be also integral,
let it be itf=U (w, z), where U is integral.

Then the number of the zeros of w' on the surface is the number of simultaneous roots
common to the two equations U (w, z) = 0,f(w, z) = Q. If u^ and/M denote the aggregates
of the terms of highest dimensions in these equations — say of dimensions X and /j. respec
tively — then Xfj. is the number of common roots, that is, the number of zeros of w1.

The number of points, where it/ assumes a value A, is the number of simultaneous
roots common to the equations U (w, z) = A, f(w, z) = Q, that is, it is X/x as before. Hence
there are as many points where vf assumes a given value as there are zeros of w'\ and
therefore the number of the infinities is the same as the number of zeros. The number
of infinities can also be obtained by considering them as simultaneous roots common to

^=0,^=0.

If the function be not integral, it can (§193) be expressed in the form w' = -„>-- -'— .- , where

V (W, Z)

U and V are integral, rational algebraical functions. The zeros of uf are the zeros of U
and the infinities of F, the numbers of which, by what precedes, are respectively the same
as the infinities of U and the zeros of V. The latter are the infinities of vf; and therefore
w' has as many zeros as it has infinities.

Note. When the numerator and the denominator of a uniform fractional
function of z have a common zero, we divide both of them by their greatest
common measure; and the point is no longer a common zero of their new
forms. But when the numerator U (w, z) and the denominator V(w, z) of a
uniform function of position on a Riemann's surface have a common zero, so
that there are simultaneous values of w and z for which both vanish, U and V
do not necessarily possess a rational common factor ; and then the common
zero cannot be removed.

It is not difficult to shew that this possibility does not affect the preceding theorem.

195. In the case of uniform functions it was seen that, as soon as theii
integrals were considered, deviations from uniformity entered. Special inves
tigations indicated the character of the deviations and the limitations
their extent. Incidentally, special classes of functions were introduced,
such as many-valued functions, the values differing by multiples of a
constant ; and thence, by inversion, simply-periodic functions were deduced.

So, too, when multiform functions denned by an algebraical equation are
considered, it is necessary to take into special account the deviations from
uniformity of value on the Riemann's surface which may be introduced by
processes of integration. It is, of course, in connexion with the branch
points that difficulties arise ; but, as the present method of representing the
variation of the variable is distinct from that adopted in the case of uniform

195.] PATHS OF INTEGRATION 373

functions, it is desirable to indicate how we deal with, not merely branch
points, but also singularities of functions when the integrals of such functions
are under consideration. In order to render the ideas familiar and to avoid
prolixity in the explanations relating to general integrals, we shall, after
one or two propositions, discuss again some of the instances given in
Chapter IX., taking the opportunity of stating general results as occasion
may arise.

One or two propositions already proved must be restated : the difference
from the earlier forms is solely in the mode of statement, and therefore the
reasoning which led to their establishment need not be repeated.

I. The path of integration between any two points on a Riemann's surface
can, without affecting the value of the integral, be deformed in any possible
continuous manner that does not make the path pass over any discontinuity of
the subject of integration.

This proposition is established in § 100.

II. A simple closed curve on a Riemann's surface, which is a path of
integration, can, without affecting the value of the integral, be deformed in
any possible continuous manner that does not make the curve pass over any
discontinuity of the subject of integration.

Since the curve on the surface is closed, the initial and the final points
are the same ; the initial branch of the function is therefore restored after
the description of the curve. This proposition is established in Corollary II.,
§100.

III. If the path of integration be a curve between two points on different
sheets, determined by the same algebraical value of z, the curve is not a closed
curve ; it must be regarded as a path betiueen the two points ; its deformation
is subject to Proposition I.

No restatement, from Chapter IX., of the value of an integral, along
a path which encloses a branch-point, is necessary. The method of dealing
with the point when that value is infinite will be the same as the method of
dealing with other infinities of the function.

196. We have already obtained some instances of multiple-valued
functions, in the few particular integrals in Chapter IX. ; the differences in
the values of the functions, arising as integrals, consist solely of multiples of
constants. The way in which these constants enter in Riemann's method is
as follows.

When the surface is simply connected, there is no substantial difference
from the previous theory for uniform functions ; we therefore proceed to the
consideration of multiply connected surfaces.

On a general surface, of any connectivity, take any two points z0 and z.
As the surface is one of multiple connection, there will be at least two

374 CONSTANT OF INTEGRATION [196.

essentially distinct paths between z0 and z, that is, paths which cannot be
reduced to one another ; one of these paths can be deformed so as to be
made equivalent to a combination of the other with some irreducible circuit.
Let z1 denote the extremity of the first path, and let z.2 denote the same point
when regarded as the extremity of the second ; then the difference of the
two paths is an irreducible circuit passing from z± to z%. When this circuit
is made impossible by a cross-cut G passing through the point z, then zl
and #2 may be regarded as points on the opposite edges of the cross-cut : and
the irreducible circuit on the unresolved surface becomes a path on the
partially resolved surface passing from one edge of the cross-cut to the other.

When the surface is resolved by means of the proper number of cross-cuts
into a simply connected surface, there is still a path in the surface from
z-i to £2 on opposite edges of the cross-cut G : and all paths between zl and
-Z2 in the resolved surface are reconcileable with one another. One such path
will be taken as the canonical path from zl to £2; it evidently does not meet
any of the cross-cuts, so that we consider only those paths which do not
intersect any cross-cut.

If then Z be the function of position on the surface to be integrated, the
value of the integral for the first path from z0 to z^ is

f

•> Z

Zdz;

e*t

and for the second path it is I Zdz,

J Z0

or, by the assigned deformation of the second path, it is

, /"Si

Zdz + Zdz,

fZ

J t

the second integral being taken along the canonical path from z1 to z2 in the
surface, that is, along the irreducible circuit of canonical form, which would be
possible in the otherwise resolved surface were the cross-cut G obliterated.
The difference of the values of the integral is evidently

/;

Zdz,

rz
which is therefore the change made in the value of the integral I Zdz,

J Zo

when the upper limit passes from one edge of the cross-cut to the other ; let
it be denoted by /. As the curve is, in general, an irreducible circuit, this
integral / may not, in general, be supposed zero.

We can arbitrarily assign the positive and the negative edges of some one
cross-cut, say A. The edges of a cross-cut B that meets A are defined to be
positive and negative as follows : when a point moves from one edge of B to
the other, by describing the positive edge of A in a direction that is to the
right of the negative edge of A, the edge of B on which the point initially

196.] AT A CROSS-CUT 375

lies is called its positive edge, and the edge of B on which the point finally
lies is called its negative edge. And so on with the cross-cuts in succession.

The lower limit of the integral determining the modulus for a cross-cut
is taken to lie on the negative edge, and the upper on the positive edge.

Kegarding a point £ on the cross-cut as defining two points ^ and z» on
opposite edges which geometrically are coincident, we now prove that for all
points on the cross-cut which can be reached from £ without passing over any
other cross-cut, when the surface is resolved into one that is simply connected,
the integral I is a constant. For, if £' be such a point, defining z/ and z2' on
opposite edges, then z1z.2z.2'2:1'21 is a circuit on the simply connected surface,
which can be made evanescent ; and it will be assumed that no infinities of Z
lie in the surface within the circuit, an assumption which will be taken into
account in §§ 197, 199. Therefore the integral of Z, taken round the circuit,
is zero. Hence

Zdz + Zdz + Zdz + Zdz = 0,

/% J Z2' Jf,'

that is, p Zdz - \ Zdz = \ Zdz - [" Zdz.

J Z, J Zi J Zi J 22

Along the direction of the cross-cut, the function Z is uniform : and
therefore Zdz is the same for each element of the two edges, so long as the
cross-cut is not met by any other. Hence the sums of the elements on the
two edges are the same for all points on the cross-cut that can be reached
from £ without meeting a new cross-cut. The two integrals on the right-
hand side of the foregoing equation are equal to one another, and therefore
also those on the left-hand side, that is,

f*2 C^

Zdz = Zdz,

J Sl * 2/

which shews that the integral I is constant for different points on a portion of
cross-cut that is not met by any other cross-cut.

If however the cross-cut be met by another cross-cut C', two cases arise
according as C" has only one extremity, or has both extremities, on C.

First, let C' have only one extremity 0 on C. By what precedes, the
integral is constant along OP, and it is constant
along OQ ; but we cannot infer that it is the same R'

constant for the two parts. The preceding proof
fails in this case ; the distance z.,z.^ in the resolved
surface is not infinitesimal, and therefore there is Q'

no element Zdz for z^zj to be the same as the Q z\ O ?j

element for z^'. Let 72 be the constant for OP, 7j Fig. 66.

that for QO ; and let QP be the negative edge. Then

,= Zdz, 1,= Zdz.

376 MODULUS OF PERIODICITY [196.

Let /' be the constant value for the cross-cut OR, and let OR be the
negative edge ; then

/•z/

/'=/ Zdz.

j ~

•> i>2

In the completely resolved surface, a possible path from z.2 to z% is z2 to z1} z^
to Zi, z^ to £/ ; it therefore is the canonical path, so that

/-I fZl f22'

Zdz + Zdz+\ Zdz
'-1 •/% J Z,'

= - /3 + /! + r ' zdz.

JM,

f*1

But I Zdz is an integral of a uniform finite function along an infinitesimal

arc z-£)z{, and it is zero in the limit when we take zy and £/ as coincident.
Thus

/' = !,-/„

or the constant for the cross-cut OR is the excess of the constant for the part of
PQ at the positive edge of OR over the constant for the part of PQ at the
negative edge.

Secondly, let G' have both extremities on G, close to one another so that
they may be brought together as in the figure : it
is effectively the case of the directions of two cross
cuts intersecting one another, say at 0. Let I1, I2,
Is, 74 be the constants for the portions QO, OP, OR,
SO of the cross-cuts respectively, and let z3z2 be
the positive edge of QOP ; then z±z3 is the positive
edge of SOR. Then if © (z) denote the value of

fz
the integral I Zdz at 0, which is definite because

J 20 •tl8' "'•

the surface is simply connected and no discontinuities of Z lie within the
paths of integration, we have

72 = r Zdz = © (>2) - <H) (^) ;

1
and /3 = !" Zdz = © (*,) - @ (z,), 74 = [" Zdz = © (z,) - © (z,} ;

J 22 J Z,

so that /! — /2 = /3 — /4,

or the excess of the constant for the portion of a cross-cut on the positive edge,
over the constant for the portion on the negative edge, of another cross-cut is
equal to the excess, similarly estimated, for that other cross-cut.

Ex. Consider the constants for the various portions of the cross-cuts in the canonical
resolution (§§ 180, 181) of a Riemann's surface. Let the constants for the two portions
of ar be Ar, Ar' ; and the constants for the two portions of br be Br, Br' ; and let the
constant for cr be Cr.

196.] FOR A CROSS-CUT

Then, at the junction of cr and ar + lt we have

at the junction of cr and br, we have

and, at the crossing of ar and br, we have A

Now, because bl is the only cross-cut which meets alt '
we have A1 = A1' ; hence Bl = Bj1, and therefore Cv = 0.
Hence A2 = A2'; therefore B^=-B^ and therefore also
<7., = 0. And so on.

377

Fig. 68.

Hence the constant for each of the portions of a cross-cut a is the same ; the constant for
each of the portions of a cross-cut b is the same ; and the constant for each cross-cut c is zero.
A single constant may thus be associated with each cross-cut a, and a single constant with
each cross-cut b, in connexion with the integral of a given uniform function of position on
the Riemann's surface. It has not been proved — and it is not necessarily the fact — that
any one of these constants is different from zero ; but it is sufficiently evident that, if all
the constants be zero, the integral is a uniform function of position on the surface, that is,
a rational algebraical function of w and z.

197. Hence the values of the integral at points on opposite edges of a
cross-cut differ by a constant.

Suppose now that the cross-cut is obliterated : the two paths to the point
z will be the same as in the case just considered and will furnish the same
values respectively, say U and U + 1. But the irreducible circuit which
contributes the value / can be described any number of times ; and
therefore, taking account solely of this irreducible circuit and of the cross-cuts
which render other circuits impossible on the resolved surface, the general
value of the integral at the point z is

U+kl,
where k is an integer and U is the value for some prescribed path.

The constant / is called* a modulus of periodicity.

It is important that every modulus of periodicity should be finite; the path
which determines the modulus can therefore pass through a point c where
Z = GO , or be deformed across it without change in the modulus, only if the
limit of (z — c) Z be a uniform zero at the point. If, however, the limit of
(z — c) Z at the point be a constant, implying a logarithmic infinity for the
integral, or if it be an infinity of finite order (the order not being necessarily
an integer), implying an algebraical infinity for the integral, we surround
the point c by a simple small curve and exclude the internal area from the
range of variation of the independent variable-}-. This exclusion is secured
by making a small loop-cut in the surface round the point; it increases
by unity the connectivity of the surface on which the variable is represented.

* Sometimes the modulus for the cross-cut.

t This is the reason for the assumption made on p. 375.

378 THE NUMBER OF INDEPENDENT MODULI [197.

When the limit of (z — c)Z is a uniform zero at c, no such exclusion
is necessary: the order of the infinity for Z is easily seen to be a proper
fraction and the point to be a branch-point.

Similarly, if the limit of zZ for z = co be riot zero and the path which
determines a modulus can be deformed so as to become infinitely large, it is
convenient to exclude the part of the surface at infinity from the range of
variation of the variable, proper account being taken of the exclusion. The
reason is that the value of the integral for a path entirely at infinity (or
for a point-path on Neumann's sphere) is not zero ; z = <x> is either a
logarithmic or an algebraic infinity of the function. But, if the limit of zZ
be zero for £ = oo , the exclusion of the part of the surface at infinity is
unnecessary.

198. When, then, the region of variation of the variable is properly
bounded, and the resolution of the surface into one that is simply connected
has been made, each cross-cut or each portion of cross-cut, that is marked off
either by the natural boundary or by termination in another cross-cut,
determines a modulus of periodicity. The various moduli, for a given
resolution, are therefore equal, in number, to the various portions of the
cross-cuts. Again, a system of cross-cuts is susceptible of great variation,
not merely as to the form of individual members of the system (which does
not affect the value of the modulus), but in their relations to one another.
The total number of cross-cuts, by which the surface can be resolved into one
that is simply connected, is a constant for the surface and is independent of
their configuration : but the number of distinct pieces, defined as above, is
not independent of the configuration. Now each piece of cross-cut furnishes
a modulus of periodicity ; a question therefore arises as to the number of
independent moduli of periodicity.

Let the connectivity of the surface be N+ 1, due regard being had to the
exclusions, if any, of individual points in the surface : in order that account
may be taken of infinite values of the variable, the surface will be assumed
spherical. The number of cross-cuts necessary to resolve it into a surface
that is simply connected is N; whatever be the number of portions of the
cross-cuts, the number of these portions is not less than N.

When a cross-cut terminates in another, the modulus for the former and
the moduli for the two portions of the latter are connected by a relation

= &),- too

so that the modulus for any portion can be expressed linearly in terms of
the modulus for the earlier portion and of the modulus for the dividing
cross-cut.

198.] IS EQUAL TO THE CONNECTIVITY 379

Similarly, when the directions of two cross-cuts intersect, the moduli of
the four portions are connected by a relation

and by passing along one or other of the cross-cuts, some relation is obtainable
between wl and &>/ or between w2 and to./, so that, again, the modulus of any
portion can be expressed linearly in terms of the modulus for the earlier
portion and of moduli independent of the intersection.

Hence it appears that a single constant must be associated with each
cross-cut as an independent modular constant ; and then the constants
for the various portions can be linearly expressed in terms of these inde
pendent constants. There are therefore N linearly independent moduli of
periodicity: but no system of moduli is unique, and any system can be
modified partially or wholly, if any number of the moduli of the system be
replaced by the same number of independent linear combinations of members
of the system. These results are the analytical equivalent of geometrical
results, which have already been proved, viz., that the number of independent
simple irreducible circuits in a complete system is N, that no complete
system of circuits is unique, and that the circuits can be replaced by
independent combinations reconcileable with them.

199. If, then, the moduli of periodicity of a function U at the cross-cuts
in a resolved surface be Il} /2, ..., IN, all the values of the function at
any point on the unresolved surface are included in the form

7T_I_ /n-t T _i_ -m T 4. _L T

where m1,m2, ..., inx are integers.

Some special examples, treated by the present method, will be useful in leading up to
the consideration of integrals of the most general functions of position on a Riemann's
surface.

Ex. 1. Consider the integral I — .

J z

The subject of integration is uniform, so that the surface is one-sheeted. The origin
is an accidental singularity and gives a logarithmic infinity for the integral ; it is therefore
excluded by a small circle round it. Moreover, the value of the integral round a circle
of infinitely large radius is not zero: and therefore 2 = 00 is excluded from the range of
variation. The boundary of the single spherical sheet can be taken to be the point
s= co ; and the bounded sheet is of connectivity 2, owing to the small circle at the origin.
The surface can be resolved into one that is simply connected by a single cross-cut drawn
from the boundary at 2—00 to the circumference of the small circle.

If a plane surface be used, this cross-cut is, in effect, a section (§ 103) of the plane
made from the origin to the point 2 = 00.

There is only one modulus of periodicity : its value is evidently / — , taken round the
origin, that is, the modulus is 2ni. Hence whenever the B

path of variation from a given point to a point z passes ::-•.-.-.-.•.•.•.•--.•.-.::.--• ^=Q °

from A to B, the value of the integral increases by 2ni ; but A

if the path pass from B to A, the value of the integral ^o- fi9-

decreases by 2ni. Thus A is the negative edge, and B the positive edge of the cross-cut.

380 EXAMPLES [199.

If, then, any one value of I - be denoted by w, all values at the point in the

J Zo z
unresolved surface are of the form w + 2mni, where in is an integer ; when z is regarded

as a function of w, it is a simply- periodic function, having 2n-t for its period.

Ex. 2. Consider I -^ ^. The subject of integration is uniform, so that the surface

consists of a single sheet. There are two infinities ±a, each of the first order, because
(z + a)Z is finite at these two points : they must be excluded by small circles. The limit,
when s= co , of z/(zz - a2) is zero, so that the point z = co does not need to be excluded. We
can thus regard one of the small circles as the boundary of the surface, which is then
doubly connected : a single cross-cut from the other circle to the boundary, that is, in
effect, a cross-cut joining the two points a and - a, resolves the surface into one that is
simply connected.

It is easy to see that the modulus of periodicity is — : that A is the negative edge and

B the positive edge of the cross-cut : and that, if w be

a value of the integral in the unresolved surface at any _tt0 B -fl

point, all the values at that point are included in the A

form Fig. 70.

where n is an integer.

Ex. 3. Consider J (a2 - z2)~* dz. The subject of integration is two- valued, so that the
surface is two-sheeted. The branch-points are +a, and oo is not a branch-point, so that
the single branch-line between the sheets may be taken as the straight line joining a
and -a. The infinities are ±a; but as (z + a) (a2-22)~i vanishes at the points, they do

not need to be excluded. As the limit of z (a2-22)~i, for 2 = 00, is not zero, we exclude
z=oo by small curves in each of the sheets.

Taking the surface in the spherical form, we assign as the boundary the small curve
round the point 2 = 00 in one of the sheets. The connectivity of the surface, through its
dependence on branch-lines and branch-points, is unity : owing to the exclusion of the point
2= co by the small curve in the other sheet, the connectivity is increased by one unit: the
surface is therefore doubly connected. A single cross-cut will resolve the surface into one
that is simply connected : and this cross-cut must pass from the boundary at 2 = 00 which
is in one sheet to the excluded point 2=00 .

Since the (single) modulus of periodicity is the value of the integral along a circuit in

the resolved surface from one edge of the cross-cut to _____ ..

the other, this circuit can be taken so that in the un- R ,-/-''" ""^

resolved surface it includes the two branch-points ; C-^~~

and then, by II. of § 195, the circuit can be deformed

until it is practically a double straight line in the upper

sheet on either side of the branch line, together with two

small circles round a and - a respectively. Let P be the

origin, practically the middle point of these straight lines. O'

Consider the branch (a2 - 22) ~ * belonging to the upper Fig . 71-

sheet. Its integral from P to a is

*32

fa

/

Jo

(a2 - 22)

From a to -a the branch is -(a2-*2)"* ; the point R is contiguous in the surface,

199.] OF MODULI OF PERIODICITY 381

not to P, but (as in § 189) to the point in the second sheet beneath P at which the branch is

— (a2-22) , the other branch having been adopted for the upper sheet. Hence, from a
to —a by R, the integral is

/_2 »2\ fJ~
— ( Or — Z I U4.

From - a to Q, the branch is + (a2 — 22) 2, the same branch as at P : hence from - a to Q,
the integral is

f° («2-22)~i^.
J -a

The integral, along the small arcs round a and round a' respectively, vanishes for each.
Hence the modulus of periodicity is

f "(a2 - z*)~*dz + f~a - (a2 - 22)-i dz + f° (a2 - z^ dz,

Jo Jo, J -a

that is, it is 2rr.

This value can be obtained otherwise thus. The modulus is the same for all points
on the cross-cut; hence its value, taken at 0' where 2 = 00, is

J(a2-22)~i^,

passing from one edge of the cross-cut at (J to the other, that is, round a curve in the
plane everywhere at infinity. This gives

2ni Lt 2(«2-22)~* = ^ = 27r,

3=OO £

the same value as before.

The latter curve round (7, from edge to edge, can easily be deformed into the former
curve round a and - a from edge to edge of the cross-cut.

Again, let w1 be a value of the integral for a point z1 in one sheet and «?2 be a value for
a point 2a in the other sheet with the same algebraical value as z1: take zero as the
common lower limit of the integral, being the same zero
for the two integrals. This zero may be taken in either
sheet, let it be in that in which zl lies : and then

Pig.72.

To pass from 0 to z2 for w2, any path can be justifiably deformed into the following:
(i) a path round either branch-point, say a, so as to return to the point under 0 in the
second sheet, say to 02, (ii) any number m of irreducible circuits round a and -a, always
returning to 02 in the second sheet, (iii) a path from 02 to 22 lying exactly under the path
from 0 to 2X for wt. The parts contributed by these paths respectively to the integral w2
are seen to be

fa, ro

(i) a quantity + TT, arising from J Q (a2 - 22) ~^ dz + j ^ - (a2 - 22)~* dz, for reasons

similar to those above ;
(ii) a quantity »i27r, where m is an integer positive or negative ;

(iii) a quantity I " - (a2 - 22)~* dz.

J 02

In the last quantity the minus sign is prefixed, because the subject of integration is
everywhere in the second sheet. Now z2 = zlt and therefore the quantity in (iii) is

- [*

that is, it is - wv ; hence w2 = (2m + 1 ) TT -

382 MODULI OF PERIODICITY [199.

If then we take ?0= / (a2 — z2)~^ dz, the integral extending along some denned curve from

J o

an assigned origin, say along a straight line, the values of w belonging to the same
algebraical value of z are 2nn + w or (2m + l)»r — w; and the inversion of the functional

relation gives

(j) (w) =z = (f> {(2mr + w)

where m and n are any integers.

Ex. 4. Consider I - r, assuming |c|>|a . The surface is two-sheeted,
j (z-o)(a*-#}

with branch-points at ±a but not at QC : hence the line joining a and -a is the sole

branch-line. The infinities of the subject of integration are a, —a, and c.. Of these a

and -a need not be excluded, for the same reason that

their exclusion was not required in the last example. But

c must be excluded ; and it must be excluded in both

sheets, because z = c makes the subject of integration

infinite in both sheets. There are thus two points of

accidental singularity of the subject of integration ; in

the vicinity of these points, the two branches of the

subject of integration are

_!_(««_<*)-*+..., _ J_(a._cS)-i_...) ' FJg- 73'

Z — C 6 C

the relation between the coefficients of (z - c)"1 in them being a special case of a more
general proposition (§ 210). And since z/{(z - c) (a2 - z2)*} when 2=00 is zero, oo does not
need to be excluded.

The surface taken plane is doubly connected, as in the last example, one of the curves
surrounding c, say that in the upper sheet, being taken as the boundary of the surface.
A single cross-cut will suffice to make it simply connected : the direction of the cross-cut
must pass from the c-curve in the lower sheet to the branch-line and thence to the
boundary in the upper sheet.

There is only a single modulus of periodicity, being the constant for the single cross-cut.
This modulus can be obtained by means of the curve AB in the first sheet; and, on
contraction of the curve (by II, § 195) so as to be infmitesimally near c, it is easily seen to be
2-Trt (a2 - c2)~*, or say 2n- (c2 - «2)~^. But the modulus can be obtained also by means of
the curve CD; and when the curve is contracted, as in the previous example, so as
practically to be a loop round a and a loop round -a, the value of the integral is

dz

[a

J-

which is easily proved to be 2n-(c2 — a2)~a.

As in Ex. 4, a curve in the upper sheet which encloses the branch-points and the
branch-lines can be deformed into the curve AB.

Ex.b. Consider w=$(4z3-g<,z-g3y~* dz=$udz.

The subject of integration is two-valued, and therefore the Riemann's surface is two-
sheeted. The branch-points are z = co , elt e%, e3 where elt e%, e3 are the roots of

4s3-#2Z-#3 = 0;
and no one of them needs to be excluded from the range of variation of the variable.

199.]

OF INTEGRALS

383

The connectivity of the surface is 3, so that two cross-cuts are necessary to resolve
the surface into one that is simply connected. The configurations of the branch-lines and
Q2

Fig. 74.

of the cross-cuts admit of some variety; two illustrations of branch-lines are given in
Fig. 74, and a point on Q: in each diagram is taken as boundary.

The modulus for the cross-cut Q1 — say from the inside to the outside — can be obtained
in two different ways. First, from P, a point on Ql, draw a line to e2 in the first sheet,
then across the branch-line, then in the second sheet to es and across the branch-line,
then in the first sheet round e3 and back to P : the circuit is represented by the double
line between e2 and ez. The value of the integral is

fea C&2 fea

I udz+ I (-u)dz, that is, 2 I udz.
J et J ea J <?2

Again, it can be obtained by a line from P', another point on Qlt to oo , round the branch
point there and across the branch-line, then in the second sheet to el and round elt then
across the branch-line and back to P : the value of the
integral is

P°
JL»-2 j udz.

J «i
But the modulus is the same for P as for P' : hence

f°° fes

= 2 udz = 2 udz.

J el J e2

This relation can be expressed in a different form. The
path from e2 to e3 can be stretched into another form
towards 2 = 00 in the first sheet, and similarly for the
path in the second sheet, without affecting the value of
the integral. Moreover as the integral is zero for 2 = 00,
we can, without affecting the value, add the small part
necessary to complete the circuits from e2 to oo and from e3 to oo .
circuits being given by the arrows, we have

Fig. 75.
The directions of these

or, if

for X = l, 2, 3, we have*

say

and El is the modulus of periodicity for the cross-cut Ql .

* See Ex. 6, § 104.

fe, r« fe,

I udz = 2 / udz + 2 I udz,
J e* J <?2 J oo

/•
ttcfe,

ps

El = 2j^udz = E2-Es,

384 EXAMPLES [199.

In the same way the modulus of periodicity for Q2 is found to be

r°° fe,

E3=2 I udz and to be 2 I udz,
J e, J <?2

the equivalence of which can be established as before.

Hence it appears that, if to be the value of the integral at any point in the surface,

the general value is of the form w + mE1 + nE3, where m and n are integers. As the
integral is zero at infinity (and for other reasons which have already appeared), it is
convenient to take the fixed limit z0 so as to define w by the relation

w= I udz.

Now corresponding to a given algebraical value of z, there are two points in the surface
and two values of w : it is important to know the relation to one another of these two
values. Let z' denote the value in the lower sheet : then the path from z' to oo can be
made up of

(i) a path from z1 to oo ' ; (ii) any number of irreducible circuits from oo ' to oo ' ; and
(iii) across the branch-line and round its extremity to oo .

These parts respectively contribute to the integral

.-QO' roc

(i) a quantity I (-u)dz, that is, -I udz, or, -w; (ii) a quantity mEl + nE3,

J z' J z

where m and n are integers ; (iii) a quantity zero, since the integral vanishes

at infinity : so that w' = mE1 + nE3-w.

If now we regard z as a function of w, say z = $> (M>), we have

But z' — z algebraically, so that we have

0 = $> (w) = jp (mE1 + 7iE3 ± u-
as the function expressing z in terms of w.
Similarly it can be proved that

the upper and the lower signs being taken together. Now g> (w), by itself, determines a
value of z, that is, it determines two points on the surface : and $' (w) has different values
for these two points. Hence a point on the surface is uniquely determined by fp(w) an

Ex. 6. Consider w = j {(1 -z2) (I - t?s?)}-* dz = J udz. The subject of integration is

two-valued, so that the surface is two-sheeted. The branch-points are ±1, ±j but
not oo ; no one of the branch-points need be excluded, nor need infinity.

The connectivity is 3, so that two cross-cuts will render the surface simply connected :
let the branch-lines and the cross-cuts be taken as in the figure.

The details of the argument follow the same course as in the previous case.

The modulus of periodicity for Q2 is 2 I udz = 4 I udz = 4K, in the ordinary

notation.

i
The modulus of periodicity for Ql is 2 / udz = 2iA", as before.

199.]

OF MODULI OF PERIODICITY

385

Hence, if w be a value of the integral for a point z in the first sheet, a more general
value for that point is w+m4K + n2iK'.

Let uf be a value of the integral for a point 2' in the second sheet, where z" is
algebraically equal to z— the point in the
first sheet at which the value of the integral
is w • then

w' = 2K + m4K -f n2iK' — w,
so that, if we invert the functional relation
and take z = snw, we have Fig. 76.

sn w = z = sn (w + 4mK+ 2niK')

= sn {(4m + 2) K+ 2niK' - w}.

Ex. 7. Consider the integral w = I- — Z—- , where u = f(I-z*)(l-

J (z—c)u

As in the last case, the surface is two-sheeted : the branch-points are ±1, + -1 but no
one of them need be excluded, nor need z = ao . But the point z=c must be excluded in
both sheets ; for expanding the subject of integration for points in the first sheet in the
vicinity of z = c, we have

1 _,

z — c
and for points in the second sheet in the vicinity of z=c, we have

1 -i

Z — 0

in each case giving rise to a logarithmic infinity for z = c.

We take the small curves excluding z = c in both sheets as the boundaries of the
surface. Then, by Ex. 4 § 178, (or because one of these curves may be regarded as a

Fig. 77.

; boundary of the surface in the last example, and the curve excluding the infinity in the
other .sheet is the equivalent of a loop-cut which (§ 161) increases the connectivity by
unity), the connectivity is 4. The cross-cuts necessary to make the surface simply
connected are three. They may be taken as in the figure ; ft is drawn from the boundary
in one sheet to a branch-line and thence round * to the boundary in the other sheet: Q,
beginning and ending at a point in ft, and ft beginning and ending at a point in ft.
The moduli of periodicity are :

for ft, the quantity (Q1 = )2W{(1-C8)(1-^C2)}-1, obtained by taking a small curve
round c in the upper sheet :

F.

ft, the quantity (fl^^^-, obtained by taking a circuit round 1

and p passing from one edge of ft to the other at F:

25

386 MODULI OF PERIODICITY [199.

Qs, the quantity (O3 = )2 f k . _* , obtained by taking a circuit round -1

and -T, passing from one edge of <?3 to the other at G:
Ic

so that, if any value of the integral at a point be w, the general value at the point is

where mn m2, m3 are integers.

Conversely, z is a triply-periodic function of w; but the function of w is not uniform
(§ 108).

Ex. 8. As a last illustration for the present, consider

The surface is two-sheeted ; its connectivity is 3, the branch-points being ± 1, ± ^ but not
z = oo . No one of the branch-points need be excluded, for the integral is finite round each
of them. To consider the integral at infinity, we substitute z=^ , and then

dz'

giving for the function at infinity an accidental singularity of the first order in each
sheet.

The point 2=00 must therefore be excluded from each sheet: but the form of w, for
infinitely large values of z, shews that the modulus for the cross-cut, which passes from
one of the points (regarded as a boundary) to the other, is zero.

The figure in Ex. 6 can be used to determine the remaining moduli. The modulus
for $ is

fi /1-&V,

I-? — r' dx
J-l\ I-*2

I

=4r — ^=^n«fe

with the notation of Jacobian elliptic functions. The modulus for Qt is

i

= 2 I ( — -f] dx

Jl\ L-X J

on transforming by the relation £%2 + £'V = l : the last expression can at once be changed
into the form 2z (K - -£")> witn the same notation as before.

If then w be any value of the integral at a point on the surface, the general value

there is

w + 4mE+ 2m (K' - E'\

where m and n are integers.

200.] INTEGRAL OF ALGEBRAIC FUNCTION 387

200. After these illustrations in connection with simple cases, we may
proceed with the consideration of the integral of the most general function
w' of position on a general Riemann surface, constructed in connection with
the algebraical equation

/ (w, z) = wn + wn~lg, (*) + ... + wgn^ (z) + gn (z) = 0,

where the functions g(z) are rational, integral and algebraical. Subsidiary
explanations, which are merely generalised from those inserted in the
preceding particular discussions, will now be taken for granted.

Taking w' in the form of § 193, we have

m> = \ *. (,) + '" (*> """' +df • + h-< (*> = 1 h, w + £ffi^ ,

dw dw

1 f
so that in taking the integral of w' we shall have a term - I h0 (z) dz, where

n j

h0 (z) is a rational algebraical function. This kind of integral has been
discussed in Chapter II.; as it has no essential importance for the present
investigation, it will be omitted, so that, without loss of generality merely
for the present purpose*, we may assume h0(z)to vanish; and then the
numerator of w' is of degree not higher than n — 2 in w.

The value of z is insufficient to specify a point on the surface : the values
of w and z must be given for this purpose, a requisite that was unnecessary
in the preceding examples because the point z was spoken of as being in the
upper or the lower of the two sheets of the various surfaces. Corresponding
to a value a of z, there will be n points : they may be taken in the form
(a1} ocj), (a2, a2), ..., (an, an), where al, ..., an are each algebraically equal to a,
and ttj, ..., an are the appropriately arranged roots of the equation

f(w, a) = 0.
The function w' to be integrated is of the form ' ' , where U is of

«/

dw

degree n — 2 in w, but though algebraical and rational it is not necessarily
integral in z.

An ordinary point of w', which is neither an infinity nor a branch-point,
is evidently an ordinary point of the integral.

The infinities of the subject of integration are of prime importance.
They are:

(i) the infinities of the numerator,
(ii) the zeros of the denominator.
The former are constituted by (a), the poles of the coefficients of powers of w

* See § 207, where h0 (z) is retained.

25—2

388 INTEGRALS [200.

in U (w, z), and (ft), z = cc: this value is included, because the only infinities
of w, as determined by the fundamental equation, arise for infinite values of
z, and infinite values of w and of z may make the numerator U(w, z)
infinite.

So far as concerns the infinities of w' which arise when z=<x> (and there
fore w = oo ), it is not proposed to investigate the general conditions that the
integral should vanish there. The test is of course that the limit, for z = oo ,

of - — y z' should vanish for each of the n values of w.

"L

dw

But the establishment of the general conditions is hardly worth the
labour involved ; it can easily be made in special cases, and it will be
rendered unnecessary for the general case by subsequent investigations.

201. The simplest of the instances, less special than the examples
already discussed, are two.

The first, which is really that of most frequent occurrence and is of very
great functional importance, is that in which / (w, z) = 0 has the form

where S (z) is of order 2m — 1 or 2m and all its roots are simple : then

7)f zU (w z}

J-='2w = ^S(z}. In order that the limit of jar*1' may be zero when

dw °J_

dw

z = oo , we see (bearing in mind that U, in the present case, is independent of
w) that the excess of the degree of the numerator of U over its denominator
may not be greater than m - 2. In particular, if U be an integral function
of z, a form of U which would leave fw'dz zero at z = oo is
U = c0zm-2 + dzm~3 + . . . + cm-3z + cm_2.

As regards the other infinities of Uj^8(z\ they are merely the roots of
S(z) = Q or they are the branch-points, each of the first order, of the
equation

By the results of § 101, the integral vanishes round each of these points ; and'
each of the points is a branch-point of the integral function. The integral is
finite everywhere on the surface: and the total number of such integrals,
essentially different from one another, is the number of arbitrary coefficients
in U, that is, it is m—l, the same as the class of the Riemann's surface
associated with the equation.

202. The other important instance is that in which the fundamental
equation is, so to speak, a generalised equation of a plane curve, so that gs (z)
is an integral algebraical function of z of degree s : then it is easy to see that,

202.] OF ALGEBRAIC FUNCTIONS 389

at z=oc , each branch w^z, so that ^«*"^*: hence U (w, z) can vary only
as 2n~s, in order that the condition may be satisfied. If then U(w, z) be an
integral function of z, it is evident that it can at most take a form which
makes U= 0 the generalised equation of a curve of degree n - 3; while, if it be
V (w, z)
z Jc > then V(w> z\ supposed integral in z, can at most take a form which

makes V= 0 the generalised equation of a curve of degree n — 2.

Other forms are easily obtainable for accidental singularities of coefficients
of w in U (w, z) that are of other orders.

As regards the other possible infinities of the integral, let c be an acci
dental singularity of a coefficient of some power of w in U(w, z) ; it may be

7}f
assumed not to be a zero of j*- . Denote the n points on the surface by

(CL kj), (c2, &2), ..., (cn, kn), where d, c2, ..., cn are algebraically equal to c.
In the vicinity of each of these points let w' be expanded: then, near (cr,kr}
we have a set of terms of the type

,

(Z -

where P(z-cr) is a converging series of positive integral powers of z-cr.
A corresponding expansion exists for every one of the n points.

The integral of w' will therefore have a logarithmic infinity at (cr, kr),
unless Alif is zero; and it will have an algebraical infinity, unless all the
coefficients A^r, ...... , Am>r are zero.

The simplest cases are

(i) that in which the integral has a logarithmic infinity but no

algebraical infinity ; and
(ii) that in which the integral has no logarithmic infinity.

For the former, w' is of the form — ^f?* and therefore in the vicinity of cr

(g-c)±-

dw

we have w' = — ±*- + P (z - cr),

z-cr

W (k c}
the value of A1>r being y ". and W is an integral function of kr, of

dh
degree not higher than n - 2. Hence

r=l r=l y_

dkr

dkr

390 INFINITIES OF THE INTEGRAL [202.

since c is the common algebraical value of the quantities c1; c2, . .., cn. Now
&J, &2, ...,kn are the roots of

an equation of degree n, while W is of degree not higher than n — 2 ; hence,
by a known theorem*,

- W(kr,c)

r^~¥~

dkr

n

so that S Alt r = Q.

r=l

The validity of the result is not affected if some of the coefficients A vanish.
But it is evident that a single coefficient A cannot be the only non-vanishing
coefficient ', and that, if all but two vanish, those two are equal and opposite.

This result applies to all those accidental singularities of coefficients of
powers of w in the numerator of w' which, being of the first order, give rise
solely to logarithmic infinities in the integral of w'. It is of great importance
in regard to moduli of periodicity of the integral.

(ii) The other simple case is that in which each of the coefficients
Al>r vanishes, so that the integral of w' has only an algebraical infinity at
the point cr, which is then an accidental singularity of order less by unity
than its order for w'.

In particular, if in the vicinity of cr, the form of w' be

the integral has an accidental singularity of the first order.
It is easy to prove that

n

Zt A2> r = 0,
r=l

so that a single coefficient A cannot be the only non-vanishing coefficient ;
but the result is of less importance than in the preceding case, for all the
moduli of periodicity of the integral at the cross-cuts for these points vanish.
And it must be remembered that in order to obtain the subject of integration
in this form, some terms have been removed in § 200, the integral of which
would give rise to infinities for either finite or infinite values of £.

It may happen that all the coefficients of powers of w in the numerator
of w' are integral functions of £. Then 2 = oo is their only accidental
singularity ; this value has already been taken into account.

* Burnside and Panton, Theory of Equations, (3rd ed.), p. 319.

203.] OF AN ALGEBRAIC FUNCTION 391

203. The remaining source of infinities of w', as giving rise to possible
infinities of the integral, is constituted by the aggregate of the zeros of

I/1

^- = 0. Such points are the simultaneous roots of the equations

In addition to the assumption already made that /= 0 is the equation of a
generalised curve of the nth order, we shall make the further assumptions
that all the singular points on it are simple, that is, such that there are only
two tangents at the point, either distinct or coincident, and that all the
branch-points are simple.

The results of § 98 may now be used. The total number of the points
given as simultaneous roots is n (n — 1) : the form of the integral in the
immediate vicinity of each of the points must be investigated.

Let (c, 7) be one of these points on the Riemann's surface, and let
(c + £, 7 + v) be any point in its immediate vicinity.

Q •£ / \

I. If — ~ — - do not vanish at the point, then (c, 7) is a branch-point
for the function w. We then have

f (w, z) = A'% + B'\r + quantities of higher dimensions,

for points in the vicinity of (c, 7), so that u«f when |£ is sufficiently small.

Then

7\f

£r- = %B'v + quantities of higher dimensions

*{»,

when £| is sufficiently small. Hence, for such values, the subject of integra
tion is a constant multiple of

U (% c) + positive integral powers of v and £

£* + powers of £ with index > \

that is, of £"*, when |£| is sufficiently small. The integral is therefore a
constant multiple of f * when £| is sufficiently small; and its value is there
fore zero round the point, which is a branch-point for the function repre
sented by the integral.

^•/* / \

II. If — ^J— vanish at the point, we have (with the assumptions
of § 98),

/ (w, z) = A£* + 2B& 4 6V + terms of the third and higher degrees ;
and there are two cases.

(i) If B* < AC, the point is not a branch-point, and we have
Cv + Bt; = £(B- -ACY- + integral powers £2, £3, . . .

392 INFINITIES OF ALGEBRAIC FUNCTION [203.

as the relation between v and £ deduced from/= 0. Then

7)f

^- = 2 (B£ + Cv) + terms of second and higher degrees

= X£" + higher powers of £.
In the vicinity of (c, 7), the subject of integration is

U (7, c) + Dv + E% + positive integral powers
\% + higher powers of £

Hence when it is integrated, the first term is — ^' log £, and the remain-

A.

ing terms are positive integral powers of £: that is, such a point is a
logarithmic infinity for the integral, unless U (7, c) vanish.

If, then, we seek integrals which have not the point for a logarithmic
infinity and we begin with U as the most general function possible, we can
prevent the point from being a logarithmic infinity by choosing among the
arbitrary constants in U a relation such that

There are S such points (§ 98); and therefore 8 relations among the
constants in the coefficients of U must be chosen, in order to prevent the
integral

I'

J

dw

from having a logarithmic infinity at these points, which are then ordinary
points of the integral.

(ii) If IP = AC, the point is a branch-point ; we have

££+ Cv = ^L^ + M^ + N^ + ...
as the relation between £ and v deduced from / = 0. In that case,

rlf

^- = 2 (B£ + Cv) + terms of the second and higher degrees

a

= Til* + powers of £ having indices > f .
In the vicinity of (c, 7), the subject of integration is

^ (7> c) + DV + Et, + higher powers

a ~ •

L% + higher powers of f
Hence when it is integrated, the first term is — 2 — ~ — - £~*, and it can be

_L/

proved that there is no logarithmic term ; the point is an infinity for the
integral, unless U (y, c) vanish.

If, however, among the arbitrary constants in U we choose a relation such
that

U (% c) = 0,

203.] TO BE INTEGRATED 393

then the numerator of the subject of integration

= Dv + E% + higher positive powers
— X'£" -f /u/£- + higher powers of £,

on substituting from the relation between v and £ derived from the funda
mental equation. The subject of integration then is

/£*+...

that is, *L+

L$

the integral of which is

A/
2 y £* + positive powers.

The integral therefore vanishes at the point : and the point is a branch-point
for the integral. It therefore follows that we can prevent the point from
being an infinity for the function by choosing among the arbitrary constants
in U a relation such that

There are K such points (§ 98): and therefore K relations among the
constants in the coefficients of U must be chosen in order to prevent the
integral from becoming infinite at these points. Each of the points is a
branch-point of the integral.

204. All the possible sources of infinite values of the subject of integra-

U(w, z)
tion w', = -p. — , have now been considered. A summary of the preceding

dw
results leads to the following conclusions relative to fw'dz :

(i) an ordinary point of w' is an ordinary point of the integral :

(ii) for infinite values of z, the integral vanishes if we assign proper
limitations to the form of U (w, z) :

(Hi} accidental singularities of the coefficients of powers of w in
U(w, z) are infinities, either algebraical or logarithmic or both
algebraical and logarithmic, of the integral :

(iv) if the coefficients of powers of w in U(w, z) have no accidental
singularities except for z = <x>, then the integral is finite for
infinite values of z (and of w) when U (w, z) is the most general
rational integral algebraical function of w and z of degree n - 3 ;
but, if the coefficients of powers of w in U (w, z) have an
accidental singularity of order p, then the integral will be finite

394 INTEGRALS [204.

for infinite values of z (and of w) when U(w, z) is the most
general rational integral algebraical function of w and z, the
degree in w being not greater than n — 2 and the dimensions
in w and z combined being not greater than n + p — 3 :

(v) those points, at which df/dw vanishes and which are not branch
points of the function, can be made ordinary points of the
integral, if we assign proper relations among the constants
occurring in U (w, z} :

(vi) those points, at which df/dw vanishes and which are branch
points of the function, can, if necessary, be made to furnish
zero values of the integral by assigning limitations to the
form of U (w, z) ; each such point is a branch-point of the
integral in any case.

These conclusions enable us to select the simplest and most important
classes of integrals of uniform functions of position on a Riemann's surface.

205. The first class consists of those integrals which do not acquire*
an infinite value at any point ; they are called integrals of the first kind^.

The integrals, considered in the preceding investigations, can give rise to
integrals of the first kind, if the numerator U (w, z) of the subject of integra
tion satisfy various conditions. The function U(w, z} must be an integral
function of dimensions not higher than n — 3 in w and z, in order that the
integral may be finite for infinite values of z and for all finite values of z
not specially connected with the equation / (w, z} = 0; for certain points
specially connected with the fundamental equation, being 8 + K in number,
the value of U (w, z) must vanish, so that there must be B + K relations
among its coefficients. But when these conditions are satisfied, then the
integral function is everywhere finite, it being remembered that certain
limitations on the nature of f (w, z) = 0 have been made.

Usually these conditions do not determine U (w, z) uniquely save as to a
constant factor ; and therefore in the most general integral of the first kind a
number of independent arbitrary constants will occur, left undetermined by
the conditions to which U is subjected. Each of these constants multiplies an
integral which, everywhere finite, is different from the other integrals so
multiplied ; and therefore the number of different integrals of the first kind
is equal to the number of arbitrary independent constants, left undetermined
in U. It is evident that any linear combination of these integrals, with

* They will be seen to be multiform functions even on the multiply connected Eiemann's
surface, and they do not therefore give rise to any violation of the theorem of § 40.

+ The German title is erster Gattung ; and similarly for the integrals of the second kind and
the third kind.

205.] OF THE FIRST KIND 395

constant coefficients, is also an integral of the first kind ; and therefore a
certain amount of modification of form among the integrals, after they have
been obtained, is possible.

The number of these integrals, linearly independent of one another, is
easily found. Because U is an integral algebraical function of w and z of
dimensions n— 3, it contains -|(n — 1) (n — 2) terms in its most general form ;
but its coefficients satisfy 8 + K relations, and these are all the relations that
they need satisfy. Hence the number of undetermined and independent
constants which it contains is

which, by § 182, is the class p of the Riemaiin's surface ; and therefore, for the
present case, the number of integrals, which are finite everywhere on the surface
and are linearly independent of one another, is equal to the class of the
Riemanns surface.

Moreover, the integral of the first kind has the same branch-points as the
function vu. Though the integral is finite everywhere on the surface, yet its
derivative w' is not so : the infinities of w' are the branch- points.

The result has been obtained on the original suppositions of § 98, which
were, that all the singular points of the generalised curve f(w, z) = 0 are
simple, that is, only two tangents (distinct or coincident) to the curve can
be drawn at each such point, and that all the branch-points are simple.
Other special cases could be similarly investigated. But it is superfluous to
carry out the investigation for a series of cases, because the result just
obtained, and the result of § 201, are merely particular instances of a general
theorem which will be proved in Chapter XVIII., viz., that, associated with
a Riemanns surface of connectivity 2^ + 1, there are p linearly independent
integrals of the first kind which are finite everywhere on the surface.

206. The functions, which thus arise out of the integral of an algebraical
function and are finite everywhere, are not uniform functions of position on
the unresolved surface. If the surface be resolved by 2p cross-cuts into one
that is simply connected, then the function is finite, continuous and uniform
everywhere in that resolved surface, which is limited by the cross-cuts as a
single boundary. But at any point on a cross-cut, the integral, at the two
points on opposite edges, has values that differ by any integral multiple of
the modulus of the function for that cross-cut (and possibly also by integral
multiples of the moduli of the function for the other cross-cuts).

Let the cross-cuts be taken as in § 181 ; and for an integral of the first
kind, say W , let the moduli of periodicity for the cross-cuts be

&>!, o)2, ..., a>p, for a1( a2, .
and wp+l, a)p+2,..., w.2p, for b1}

396 INTEGRALS [206.

respectively ; the moduli for the portions of cross-cuts c2, cs, ..., cp have been
proved to be zero.

Some of these moduli may vanish ; but it will be proved later (§ 231) that
all the moduli for the cross-cuts a, or all the moduli for the cross-cuts b, cannot
vanish unless the integral is a mere constant. In the general case, with which
we are concerned, we may assume that they do not vanish ; and so it follows
that, if W be a value of an integral of the first kind at any point on the
Riemanns surface, all its values at that point are of the form

Zp

W+ 2 mra)r,

r=l

where the coefficients in are integers.

The foregoing functions, arising through integrals that are finite every
where on the surface, will be found the most important from the point of
view of Abelian transcendents : but other classes arise, having infinities on
the surface, and it is important to indicate their general nature before passing
to the proof of the Existence-Theorem.

207. First, consider an integral which has algebraical, but not logarithmic,
infinities. Taking the subject of integration, as in the preceding case, to be
the most general possible, so that arbitrary coefficients enter, we can, by
assigning suitable relations among these coefficients, prevent any of the

7)f
points, given as zeros of ~- = 0, from being infinities of the integral. It

follows that then the only infinities of the integral will be the points that are
accidental singularities of coefficients of powers of w in the numerator of the
general expression for w'. These singularities must each be of the second
order at least : and, in the expansion of w' in the vicinity of each of them,
there must be no term of index — 1, the index that leads, on integration, to a
logarithm.

Such integrals are called integrals of the second kind.

The simplest integral of the second kind has an infinity for only a single
point on the surface, and the infinity is of the first order only : the integral
is then called an elementary integral of the second kind. After what has
been proved in § 202 (ii), it is evident that an elementary integral of the
second kind cannot occur in connection with the equation f(w, z) = 0, unless
the term h0 (z) of § 200 be retained in the expression for w'.

Ex. 1. Adopting the subject of integration obtained in § 200, we have

, 1; , , (7(w, z)

~n o(&) 87~'

dw

where U is of the character considered in the preceding sections, viz., it is of degree n - 2
in w ; various forms of w' lead to various forms of h0 (z) and of U (w, z).

207.] OF THE SECOND KIND 397

If -h0(z)=.- — , and if c be not a singularity of the coefficient of any power of w

iv (^2 0)

in U, it is then evident that

U(w,

mo

and the integral on the right-hand side can by choice among the constants be made an
integral of the first kind. The integral is not, however, an elementary integral of the
second kind, because z — c is an infinity in each sheet.

Ex. 2. A special integral of the second kind occurs, when we take an accidental
singularity, say z = c, of the coefficient of some power of w in U(w, z) and we neglect h0(z);
so that, in effect, the subject of integration w' is limited to the form

U being of degree not higher than n - 2 in 10. To the value z = c, there correspond n points
in the various sheets ; if, in the immediate vicinity of any one of the points, vf be of the
form

in that vicinity the integral is of the form

z-cr

For such an integral the sum of the coefficients Ar is zero : the simplest case arises
when all but two, say Al and A2, of these vanish. The integral is then of the form

A

in the vicinity of ct , and of the form

-A

^ + P.2(z-c2)
in the vicinity of c2. But the integral is not an elementary integral of the second kind.

208. To find the general value of an integral of the second kind,
all the algebraically infinite points would be excluded from the Riemann's
surface by small curves : and the surface would be resolved into one that is
simply connected. The cross-cuts necessary for this purpose would consist of
the set of 2p cross-cuts, necessary to resolve the surface as for an integral of
the first kind, and of the k additional cross-cuts in relation with the curves
excluding the algebraically infinite points.

Let the moduli for the former cross-cuts be

ej, 62, ..., ep, for the cuts alt a2, ..., ap,

€P+I> *p+i, •••, e2p for the cuts b1} b.,, ..., bp, respectively:

the moduli for the cuts c are zero. It is evident from the form of the

integral in the vicinity of any infinite point that, as the integral has only an

398 ELEMENTARY INTEGRAL [208.

algebraical infinity, the modulus for each of the k cross-cuts, obtained by a
curve from one edge to the other round the point, is zero. Hence if one
value of the integral of the second kind at a point on the surface be E (z),
all its values at that point are included in the form

Zp

E(z)+ 2 nrer,

r=l

where nt, n2, ..., n2p are integers.

The importance of the elementary integral of the second kind, inde
pendently of its simplicity, is that it is determined by its infinity, save as to an
additive integral of the first kind.

Let EI (z) and E2 (z) be two elementary integrals of the second kind,
having their single infinity common, and let a be the value of z at this point ;
then in its vicinity we have

^ <*) - ,~i + I&- «)• ^ (*) - ~:a + p> (* - a>'

and therefore A1E2(z)- AJE^(z) is finite at z = a. This new function is
therefore finite over the whole Riemann's surface : hence it is an integral of
the first kind, the moduli of periodicity of which depend upon those of El (z)
and E, (z}.

Ex. It may similarly be proved that for the special case in Ex. 2, § 207, when the
integral of the second kind has two simple infinities for the same algebraical value of z in
different sheets, the integral is determinate save as to an additive integral of the first kind.

Let a^ and «2 be the two points for the algebraical value a of z ; and let F(z) and G (z)
be two integrals of the second kind above indicated having simple infinities at ax and a2
and nowhere else.

Then in the vicinity of a: we have

F(z] = — + P1 (z - a,\ G (z) =

—

s— 0*1

so that BF(z) -AG(z) is finite in the vicinity of a^
Again, in the vicinity of «2, we have, by § 202,

so that BF(z)-AG(z) is finite in the vicinity of «2 also. Hence BF(z)- AG(z) is finite
over the whole surface, and it is therefore an integral of the first kind ; which proves the
statement.

It therefore appears that, if F (z) be any such integral, every other integral of the same
nature at those points is of the form F(z)+ W, where W is an integral of the first kind.
Now there are p linearly independent integrals of the first kind : it therefore follows that
there are p + 1 linearly independent integrals of the second kind, having simple infinities
with equal and opposite residues at two points, (and at only two points), determined by
one algebraical value of z.

208.] OF THE SECOND KIND 399

From the property that an elementary integral of the second kind is
determined by its infinity save as to an additive integral of the first kind, we
infer that there are p + l linearly independent elementary integrals of the
second kind with the same single infinity on the Riemanns surface.

This result can be established in connection with f(w, z) = 0 as follows. The subject
of integration is

\& — i*r- ^ —

' aw

where for simplicity it is assumed that a is neither a branch-point of the function
nor a singular point of the curve f(w, z) = 0, and in the present case U is of degree
n-l in w. To ensure that the integral vanishes for 3 = 00, the dimensions of U(w,z)
may not be greater than n-l. Hence U(w, z), in its most general form, is an integral,
rational, algebraical function of w and z of degree n-l; the total number of terms is
therefore £»(« + !), which is also the total number of arbitrary constants.

In order that the integral may not be infinite at each of the S + K singularities of the
curve /(w, z) = 0, a relation U(y, c) = 0 must be satisfied at each of them; hence, on this
score, there are S -f AC relations among the arbitrary constants.

Let the points on the surface given by the algebraical value a of z be (<%, ni), («2, a2),
..., (an, an). The integral is to be infinite at only one of them ; so that we must have

U(ar, ar) = 0,

for r=2, 3, ...,n; and n-l is the greatest number of such points for which U can vanish,
unless it vanish for all, and then there would be no algebraical infinity. Hence, on this
score, there are n-l relations among the arbitrary constants in U.

In the vicinity of z = a, w = a, let

then we have Q = v ?-+(?£+...

da da

where ^ is the value of £ and |£ that of |", for z- a and 10 = 0. For sufficiently small
values of |«| and |f |, we may take

**£«£

da * da
t or such points we have

U(wtz)=U(a,a)+v™+c¥+...
oa oa

and

f

Then unless ^_ LC/L?) = 1 '

df 0 (a, a) ""'"'
da

_ =

U (a, a) 3 (a, a) 8/ 8 (a, a)

8a

for («„„,), and

3 (a, a)

400 INTEGRALS [208.

for («2, a2), («3, «3), ..., («„, an), there will be terms in - in the expansion of the subject of

integration in the vicinity of the respective points, and consequently there will be
logarithmic infinities in the integral. Such infinities are to be excluded ; and therefore
their coefficients, being the residues, must vanish, so that, on this score, there appear to
be n relations among the arbitrary constants in U. But, as in § 210, the sum of the
residues for any point is zero : and therefore, when n - 1 of them vanish, the remaining
residue also vanishes. Hence, from this caxxse, there are only n — 1 relations among the
arbitrary constants in U.

The tale of independent arbitrary constants in U (w, z), remaining after all the
conditions are satisfied, is

4w(n+l)-(8 + K)-(w-l) -(n-l)

as each constant determines an integral, the inference is that there are p + l linearly
independent elementary integrals of the second kind with a common infinity.

209. Next, consider integrals which have logarithmic infinities, inde
pendently of or as well as algebraical infinities. They are called integrals of
the third kind. As in the case of integrals of the first kind and the second
kind, we take the subject of integration to be as general as possible so that it
contains arbitrary coefficients ; and we assign suitable relations among the
coefficients to prevent any of the points, given as zeros of dfjdw, from becoming
infinities of the integral. It follows that the only infinities of the integral
are accidental singularities of coefficients of powers of w in the numerator
of the general expression for w' ; and that, when w' is expanded for points in
the immediate vicinity of such an expression, the term with index — 1 must
occur.

To find the general value of an integral of the third kind, we should
first exclude from the Kiemaim's surface all the infinite points, say

Li, 1/2, ... , Ifj.,

by small curves ; the surface would then be resolved into one that is simply
connected. The cross-cuts necessary for this purpose would consist of the
set of 2p cross-cuts, necessary to resolve the surface for an integral of the
first kind, and of the additional cross-cuts, /JL in number and drawn from the
boundary (taken at some ordinary point of the integral) to the small curves
that surround the infinities of the function.

The moduli for the former set may be denoted by

CTJ , CTO , . . . , vrp for the cuts aly a2 , . . . , ap ,
and vrp+1, -n-p+2, ..., vT2p for the cuts b1} b», ..., bp respectively;

they are zero for the cuts c. Taking the integral from one edge to the other
of any one of the remaining cross-cuts llt L, ..., lq, (where lq is the cross-cut
drawn from the curve surrounding lq to the boundary), its value is given by

209.] OF THE THIRD KIND 401

the value of the integral round the small curve and therefore it is 2-TriX,
where the expansion of the subject of integration in the immediate vicinity

of z = lq is

Then, if II be any value of the integral of the third kind at a point on the
unresolved Riemann's surface, all its values at the point are included in the
form

^

11+ 2 m/nv + ZTTI £ nq\q,

r=l 7=1

where the coefficients m1, ..., m2p, «1} ..., WM are integers.

210. It can be proved that the quantities \q are subject to the relation

Let the surface be resolved by the complete system of 2p + //, cross-cuts : the
resolved surface is simply connected and has only a single boundary. The
subject of integration, w', is uniform and continuous over this resolved surface:
it has no infinities in the surface, for its infinities have been excluded ; hence

fw'dz = 0,

when the integral is taken round the complete boundary of the resolved
surface.

This boundary consists of the double edges of the cross-cuts a, b, c, L,
and the small curves round the //, points I ; the two edges of the same cross
cut being described in opposite directions in every instance.

Since the integral is zero and the function is finite everywhere along the
boundary, the parts contributed by the portions of the boundary may be con
sidered separately.

First, for any cross-cut, say aq : let 0 be the point where it is crossed by bq,
and let the positive direction of description of the whole boundary be indicated
by the arrows (fig. 81, p. 438). Then, for the portion Ca...E, the part of the

rE

integral is I w'dz, or, if Ca. . .E be the negative edge (as in § 196), the part of

w C

the integral may be denoted by

/w'dz.
c

The part of the integral for the portion F...aD, being the positive

rD (-F

edge of the cross-cut, is I w'dz, which may be denoted by - 1 w'dz. The

J F J D

course and the range for the latter part are the same as those for the
F- 26

402 ELEMENTARY INTEGRAL [210.

former, and w' is the same on the two edges of the cross-cut ; hence the

sum of the two is

a
= \ (wf — w') dz,

J c

which evidently vanishes*. Hence the part contributed to fw'dz by the two
edges of the cross-cut aq is zero.

Similarly for each of the other cross-cuts a, and for each of the cross-cuts
b, c, L.

The part contributed to the integral taken along the small curve enclosing
lq is 2Tri\q> for q = 1, 2, . . . , /A : hence the sum of the parts contributed to the
integral by all these small curves is

All the other parts vanish, and the integral itself vanishes ; hence

M

'1 ^Q = 0,

3=1

establishing the result enunciated.

COROLLARY. An integral of the third kind, that is, having logarithmic
infinities on a Riemanns surface, must have at least two logarithmic infinities.

If it had only one logarithmic infinity, the result just proved would
require that \ should vanish, and the infinity would then be purely
algebraical.

211. The simplest instance is that in which there are only two
logarithmic infinities ; their constants are connected by the equation

A.J + X2 = 0.

If, in addition, the infinities be purely logarithmic, so that there are no
algebraically infinite terms in the expansion of the integral in the vicinity
of either of the points, the integral is then called an elementary integral
of the third kind. If two points C^ and C2 on the surface be the two infini
ties, and if they be denoted by assigning the values Cj and c2 to z ; and if
Xj = 1 = — X2 (as may be assumed, for the assumption only implies division
of the integral by a constant factor), the expansion of the subject of inte
gration for points in the vicinity of Ci is

1

z — Cj

* It vanishes from two independent causes, first through the factor w'-w', and secondly
because z =zn, the breadth of any cross-cut being infinitesimal.

E C

The same result holds for each of the cross-cuts a and 6.

For each of the cross-cuts c and L, the sum of the parts contributed by opposite edges vanishes
only on account of the factor w' - w' ; in these cases the variable z is not the same for the upper
and lower limit of the integral.

211.] OF THE THIRD KIND 403

and for points in the vicinity of ca the expansion is

-1

z-c»

P2(z-c2).

Such an integral may be denoted by I112 : its modulus, consequent on
the logarithmic infinity, is

Ex. 1. Prove that, if n12, n23, H31 be three elementary integrals of the third kind
having clt c.2; c2, c3; c3, ct for their respective pairs of points of logarithmic discontinuity,
then n,2 + n23 + n31 is either an integral of the first kind or a constant.

Clebsch and Gordan pass from this result to a limit in which the points ct and c2
coincide and obtain an expression for an elementary integral of the second kind in the
form of the derivative of H13 with regard to ct. Klein, following Riemann, passes from an
elementary integral of the second kind to an elementary integral of the third kind by
integrating the former with regard to its parametric point*.

Ex. 2. Reverting again to the integrals connected with the algebraical equation
/(«-, z)=Q, when it can be interpreted as the equation of a generalised curve, an integral of
the third kind arises when the subject of integration is

where V(w, z) is of degree n- 2 in w. If V(w, z) be of degree in z not higher than n- 2,
the integral of w' is not infinite for infinite values of z; so that V(w, z) is a general integral
algebraical function of w of degree n — 2.

Corresponding to the algebraical value c of z, there are n points on the surface, say
(cn ^i)> (C2> ^2)? •••> (cni #n); and the expansion of w' in the vicinity of (cri &,.) is

the coefficients of the infinite terms being subject to the relation

V(kr,cr)

because V(w, z) is only of degree n - 2 in w. The integral of w' will have a logarithmic
infinity at each point, unless the corresponding coefficient vanish.

Not more than n- 2 of these coefficients can be made to vanish, unless they all vanish;
and then the integral has no logarithmic infinity. Let n - 2 relations, say

V(kr, cr) = 0

for r = 2, 3, ..., n, be chosen ; and let the S + K relations be satisfied which secure that the
integral is finite at the singularities of the curve / (w, z-) = 0. Then the integral is an
elementary integral of the third kind, having (cn ^) and (c2, £2) for its points of
logarithmic discontinuity.

Ex. 3. Prove that there are p + l linearly independent elementary integrals of the
third kind, having the same logarithmic infinities on the surface.

* Clebsch und Gordan, (I.e., p. 361, note), pp. 28—33 ; Klein-Fricke, Vorlesungen iiber die
Theorie der elliptischen Modulfunctionen, t. i, pp. 518—522; Biemann, p. 100.

26—2

404 CLASSES OF FUNCTIONS [211.

Ex. 4. Shew that, in connection with the fundamental equation

any integral of the first kind is a constant multiple of

[dz

Jw2'

that an integral of the second kind, of the class considered in Ex. 2, § 207, is given by

'\—w -,

and that an elementary integral of the third kind is given by

—^ dz.

Ex. 5. An elementary (Jacobian) elliptic integral of the third kind occurs in Ex. 7,
p. 385 ; and a (Jacobian) elliptic integral of the second kind occurs in Ex. 8, p. 386.

Shew that an elementary (elliptic) integral of the second kind, associated with the

equation

v*=4e*-gf-ffd>

and having its infinity at (c1? yj, is

7i (w

f
J

and that an elementary (elliptic) integral of the third kind, associated with the same
equation and having its two infinities at (c1; yj), (c2, y2)> ^s

A sufficient number of particular examples, and also of examples with'
a limited generality, have been adduced to indicate some of the properties
of functions arising, in the first instance, as integrals of multiform functions
of a variable z (or as integrals of uniform functions of position on a
Biemann's surface). The succeeding investigation establishes, from the most
general point of view, the existence of such functions on a Riemann's
surface : they will no longer be regarded as defined by integrals of multi
form functions.

CHAPTER XVII.

SCHWARZ'S PROOF OF THE EXISTENCE-THEOREM.

212. THE investigations in the preceding chapter were based on
the supposition that a fundamental equation was given, the appropriate
Riemann's surface being associated with it. The general expression of
uniform functions of position on the surface was constructed, and the
integrals of such functions were considered. These integrals in general
were multiform on the surface, the deviation from uniformity consisting
in the property that the difference between any two of the infinite number of
values could be expressed as a linear combination of integral multiples of
certain constants associated with the function. Infinities of the functions
defined by the integrals, and the classification of the functions according to
their infinities, were also considered.

But all these investigations were made either in connection with
very particular forms of the fundamental equation, or with a form of not
unlimited generality : and, for the latter case, assumptions were made,
justified by the analysis so far as it was carried, but not established generally.

In order to render the consideration of the propositions complete, it must
be inade without any limitations upon the general form of fundamental
equation.

Moreover, the second question of § 192, viz., the existence of functions
(both uniform and multiform) of position on a surface given independently of
any algebraical equation, is as yet unconsidered.

The two questions, in their generality, can be treated together. In the
former case, with the fundamental equation there is associated a Riemann's
surface, the branching of which is determined by that fundamental equation ;
in the latter case, the Riemann's surface with assigned branching is supposed

406 INITIAL SIMPLIFICATION [212.

given*. We shall take the surface as having one boundary and being other
wise closed ; the connectivity is therefore an uneven integer, and it will be
denoted by 2p + 1.

213. The problem can be limited initially, so as to prevent unnecessary
complications. All the functions to be discussed, whether they be algebraical
functions or integrals of algebraical functions, can be expressed in the form
u + iv, where u and v are two real functions of two independent real variables
x and y. It has already (§ 10) been proved that both u and v satisfy the
equation

and that, if either u or v be known, the other can be derived by a quadra
ture at most, and is determinate save as to an additive arbitrary constant.
Since therefore w is determined by u, save as to an additive constant, we
shall, in the first place, consider the properties of the real function u only.

The result is valid so long as v can be determined, that is, so long as the
function u has differential coefficients. It will appear, in the course of the
present chapter, that no conditions are attached to the derivatives of u along
the boundary of an area, so that the determination of v along such a boundary
seems open to question.

It has been (§ 36) proved, in a theorem due to Schwarz, that, if w a
function of z be defined for a half-plane and if it have real finite continuous
values along any portion of the axis of x, it can be symmetrically continued
across that portion of the axis. The continuation is therefore possible for the
real part u of the function w ; and the values of u are the real finite continuous
values of w along that portion of the axis.

It will be seen, in Chapters XIX., XX. that, by changing the independent
variables, the axis of x can be changed into a circle or other analytical line
(| 221) ; so that a function u, defined for an interior and having real finite con
tinuous values along any portion of the boundary, can be continued across that
portion of the boundary, which is therefore not the limit of existence "f of u.

* The surface is supposed given ; we are not concerned with the quite distinct question as to
how far a Kiemann's surface is determinate by the assignment of its number of sheets, its
branch-points (and consequently of its connectivity), and of its branch-lines. This question is
discussed by Hurwitz, Math. Ann., t. xxxix, (1891), pp. 1 — 61. He shews that, if ft denote the
ramification (§ 179) of the surface which, necessarily an even integer, is denned as the sum of
the orders of its branch-points, a two- sheeted surface is made uniquely determinate by assigned
branch-points; the number of essentially distinct three-sheeted surfaces, made determinate by
assigned branch-points, is ^(3n~2-l); and so on. It is easy to verify that the number of
distinct three-sheeted surfaces, with 4 assigned points as simple branch-points, is 4 : an example
suggested to me by Mr Burnside.

t The continuation indicated will be carried out for the present case by means of the com
bination of areas (§ 222), and without further reference to the transformation indicated or to
Schwarz's theorem on symmetrical continuation.

213.] POTENTIAL FUNCTION 407

The derivatives of u can then be obtained in the extended space and so v can
be determined for the boundary*.

And, what is more important, it will be found that, under conditions to be
assigned, the number of functions u that are determined is double the number
of functions w that are determined ; the complete set of functions u lead to all
the parts u and v of the functions w (§ 234, note).

214. The infinities of u at any point are given by the real parts of the
terms which indicate the infinities of w. Conversely, when the infinities of u
are assigned in functional form, those of w can be deduced, the form of the
associated infinities of v first being constructed by quadratures.

The periods of w, being the moduli at the cross-cuts, lead to real constants
as differences of u at opposite edges of cross-cuts, or, if we choose, as constant
differences of values of u at points on definite curves, conveniently taken for
reference as lines of possible cross-cuts. Conversely, a real constant modulus
for u is the real part"f* of the corresponding modulus of w.

Hence a function, w, of position on a Riemann's surface is, except as to an

additive constant, determined by a real function u of x and y (where x + iy is

the independent variable for the surface), if u be subject to the conditions : —

(i) it satisfies the equation V2u = 0 at all points on the surface where

its derivatives are not infinite :

(ii) if it be multiform, its values at any point on the surface differ by
linear combinations of integral multiples of real constants : otherwise, it is
uniform :

(iii) it may have specified infinities, of given form in the vicinity of
assigned points on the surface.

In addition to these general conditions imposed upon the function u, it is
convenient to admit as a further possible condition, for portions of the surface,
that the function u shall assume, along a closed curve, values which are
always finite. But it must be understood that this condition is used only for
subsidiary purposes : it will be seen that it causes no limitation on the final
result, all that is essential in its limitations being merged in the three
dominant conditions.

The questions for discussion are therefore (i), the existence of functions J
satisfying the above conditions in connection with a given Riemann's

* See Phragmen, Acta Math., t. xiv, (1890), pp. 225—227, for some remarks upon this
question.

t The imaginary parts of the moduli of w are determinate with the imaginary part of w : see
remark at end of § 213, and the further reference there given.

£ The functions u (and also v) are of great importance in mathematical physics for two-
dimensional phenomena in branches such as gravitational attraction, electricity, hydrodynamics
and heat. In all of them, the function represents a potential ; and, consequently, in the general
theory of functions, it is often called a potential function.

408 METHODS OF SOLUTION [214.

surface, the connectivity of which is 2p + 1 as dependent upon its branching
and the number of its sheets; and (ii), assuming that the functions exist,
their determination by the assigned conditions.

215. There are many methods for the discussion of these questions. The
potential function, both for two and for three dimensions in space, first arose in
investigations connected with mathematical physics : and, so far as concerns
such subjects, its theory was developed by Poisson, Green, Gauss, Stokes,
Thomson, Maxwell and others. Their investigations have reference to appli
cations to mathematical physics, and they do not tend towards the solution of
the questions just propounded in relation to the general theory of functions.

Klein uses considerations drawn from mathematical and experimental
physics to establish the existence of potential functions under the assigned
conditions. The proof that will be adopted brings the stages of the investi
gation into closer relations with the preceding and the succeeding parts of the
subject than is possible if Klein's method be followed*.

To establish the existence of the functions under the assigned conditions,
Riemann*f uses the so-called Dirichlet's Principle J ; but as Riemann's proof
of the principle is inadequate, his proof of the existence-theorem cannot be
considered complete.

There are two other principal, and independent, methods of importance,
each of which effectively establishes the existence of the functions, due to
Neumann and to Schwarz respectively ; each of them avowedly dispenses^
with the use of Dirichlet's Principle.

The courses of the methods have considerable similarity. Both begin
with the construction of the function for a circular area. Neumann uses
what is commonly called the method of the arithmetic mean, for gradual
approximation to the value of the potential function for a region bounded
by a convex curve : Schwarz uses the method of conformal representation,
to deduce from results previously obtained, the potential function for
regions bounded by analytical curves ; and both authors use certain
methods for combination of areas, for each of which the potential function
has been constructed ||.

* Klein's proof occurs in his tract, already quoted, Ueber Riemann's Theorie der algebraischen
Functionen und Hirer Integrate, (Leipzig, Teubner, 1882), and it is modified in his memoir "Neue
Beitrage zur Eiemann'schen Functionentheorie, " Math. Ann., t. xxi, (1883), pp. 141 — 218,
particularly pp. 160 — 162.

t Ges. Werke, pp. 35—39, pp. 96—98.

J Eiemann enunciates it, (I.e.), pp. 34, 92.

§ Neumann, Vorlesungen liber Riemann's Theorie der Abel'schen Integrale, (2nd ed., 1884),
p. 238; Schwarz, Ges. Werke, ii, p. 171.

|| Neumann's investigations are contained in various memoirs, Math. Ann., t. iii, (1871),
pp. 325—349; ib., t. xi, (1877), pp. 558—566; ib., t. xiii, (1878), pp. 255—300; ib., t. xvi,
(1880), pp. 409 — 431 ; and the methods are developed in detail and amplified in his treatise

215.] SUMMARY OF SCHWARZ'S ARGUMENT 409

What follows in the present chapter is based upon Schwarz's investi
gations : the next chapter is based upon the investigations of both Schwarz
and Neumann, and, of course, upon Riemann's memoirs.

The following summary of the general argument will serve to indicate the main line of
the proof of the establishment of potential functions satisfying assigned conditions.

I. A potential function u is uniquely determined by the conditions : that it, as

du du d2u c)2u , , . , . „ ., , . „„ -. , ,, i

well as its derivatives »-, ~-, 5-3, ^— 2- (which satisfy the equation V%=0), shall be

uniform, finite and continuous, for all points within the area of a circle ; and that, along
the circumference of the circle, the function shall assume assigned values that are always
finite, uniform and, except at a limited number of isolated points where there is a sudden
(finite) change of value, continuous. (§§ 216 — 220.)

II. By using the principle of conformal representation, areas bounded by curves other
than circles — say by analytical curves — are obtained, over which the potential function is
uniquely determined by general conditions within the area and assigned values along its
boundary. (§ 221.)

III. The method of combination of areas, dependent upon an alternating process,
leads to the result that a function exists for a given region, satisfying the general conditions
in that region and acquiring assigned finite values along the boundary, when the region
can be obtained by combinations of areas that can be conformally represented upon the
area of a circle. (§ 222.)

IV. The theorem is still valid when the region (supposed simply connected) contains
a branch-point ; the winding-surface is transformed by a relation

z-c = RZm
into a single-sheeted surface, for which the theorem has already been established.

When the surface is multiply connected, we resolve it by cross-cuts into one that is
simply connected, before discussing the function. (§ 223.)

V. Real functions exist on a Riemann's surface, which are everywhere finite and

Ueber das logarithmische und Newton'sche Potential (Leipzig, Teubner, 1877) and in his treatise
quoted in the preceding note. In this connection, as well as in relation to Schwarz's investi
gations, and also in view of some independence of treatment, Harnack's treatise, Die Grundlagen
der Theorie des logarithmischen Potentiates und der eindeutigen Potentialfunction in der Ebene
(Leipzig, Teubner, 1887), and a memoir by Harnack, Math. Ann., t. xxxv, (1890), pp. 19—40,
may be consulted.

A modification of Neumann's proof, due to Klein, is given in the first volume (pp. 508 — 522)
of the treatise cited on p. 403, note.

Schwarz's investigations are contained in various memoirs occurring in the second volume
of his Gcsammclte Werke, pp. 108—132, 133—143, 144—171, 175—210, 303—306 : their various
dates and places of publication are there stated. A simple and interesting general statement
of the gist of his results will be found in a critical notice of the two volumes of his collected
works, written by Henrici in Nature (Feb. 5, 12, 1891, pp. 321—323, 349—352). There is a
comprehensive memoir by Ascoli, based upon Schwarz's method, " Integration der Differential-
gleichung V2w = 0 in einer beliebigen Riemann'schen Fla'che," (Bih. t. kongl. Svenska Vet. Akad.
Handl., bd. xiii, 1887, Afd. 1, n. 2 ; 83 pp.) ; a thesis by Jules Riemann, Sur le probleme de
Dirichlet, (These, Gauthier-Villars, Paris, 1888), discusses a number of Schwarz's theorems
(see, however, Schwarz, Ges. Werke, t. ii, pp. 356—358) ; and an independent memoir by Prym,
Crelle, t. Ixxiii, (1871), pp. 340—364, may be consulted.

The literature of this part of the subject is very wide in extent : many other references are
given by the authors already quoted.

410 PRELIMINARY LEMMAS [215.

uniquely determinate by arbitrarily assigned real moduli of periodicity at the cross-cuts.
(§§ 224—227.)

VI. Functions exist, satisfying the conditions in (V) except that they may have at
isolated points on the surface, infinities of an assigned form. (§ 229.)

216. We shall, in the first place, treat of potential functions that have
110 infinities, either algebraical or logarithmic, over some continuous area on
the surface limited by a simple closed boundary, or by a number of non-inter
secting simple closed curves constituting the boundary ; for the present, the
area thus enclosed will be supposed to lie in one and the same sheet, so that
we may regard the area as lying in a simple plane.

At all points within the area and on its boundary, the function u is finite
and will be supposed uniform and continuous ; for all points within the area
(but not necessarily for points on the boundary), the derivatives

du du d'2u d2u
dx' dy ' dx* ' dy'2

are uniform, finite arid continuous and they satisfy the equation V2it = 0.
These may be called the general conditions.

Two cases occur according as the character of the derivatives at points in
the area is or is not assigned for points on the boundary ; if the character be
assigned, there will then be what may be called boundary conditions. The
two cases therefore are :

(A) When a function u is required to satisfy the general conditions,
and its derivatives are required to satisfy the boundary conditions :

(B) When the only requirement is that the function shall satisfy the
general conditions.

Before proceeding to the establishment of what is the fundamental
proposition in Schwarz's method, it is convenient to prove three lemmas
and to deduce some inferences that will be useful.

LEMMA I. If two functions u± and u2 satisfy the general conditions for two
regions T: and T2 respectively, which have a common portion T that is more than
a point or a line, and if ut and u2 be the same for the common portion T, then
they define a single function for the whole region composed of T: and T2.

This proposition can be made to depend upon the continuation of
analytical functions*, whether in a plane (§ 34) or, in view of a subsequent
transformation (§ 223), on a Biemann's surface.

The real function ur defines a function Wj_ of the complex variable z, for any
point in the region T^ ; and for points within this region, the function w1 is
uniquely determined by means of its own value and the values of its deriva
tives at any point within T1} obtained, if necessary, by a succession of elements
* For other proofs, see Schwarz, ii, pp. 201, 202 and references there given.

216.] FOR SCHWARZ'S PROOF 411

in continuation. Hence the value of wl and its derivatives at any point
within T defines a function existing over the whole of T^.

Similarly the real function u2 defines a function w.2 within T2, and this
function is uniquely determined over the whole of T2 by its value and the
value of its derivatives at any point within T.

Now the values of u^ and u2 are the same at all points in T, and therefore
the values of wl and w2 are the same at all points in T, except possibly for an
additive (imaginary) constant, say ia, so that

w-i = w2 + ia.

Hence for all points in T, (supposed not to be a point, so that we may have
derivatives in every direction (§ 8) : and not to be a line, so that we may
have derivatives in all directions from a point on the line), the derivatives
of wl agree with those of w.2 ; and therefore the quantities necessary to define
the continuation of wl from T over Tj agree with the quantities necessary to
define the continuation of w2 from T over T2, except only that wl and w2
differ by an imaginary constant. Hence, having regard to the form of the
elements, w{ and w.2 can be continued over the region composed of Tl and T2,
and their values differ (possibly) by the imaginary constant. When we take
the real parts of the functions, we have ut and u2 defining a single function
existing over the whole region occupied by the combination of T± and T2.

The other two lemmas relate to integrals connected with potential
functions.

LEMMA II. Let u be a function required to satisfy the general conditions,
and let its derivatives be required to satisfy the boundary conditions, for an
area S bounded by simple non-intersecting curves : then

du , ..
5- ds = 0 :
on

where the integral is extended round the whole boundary in the direction that is
positive with regard to the bounded area 8 ; and dn is an element of the normal
to a boundary-line drawn towards the interior of the space enclosed by that
boundary-line regarded merely as a simple closed curve*.

Let P and Q be any two functions, which, as well as their first and second
derivatives with regard to # and to y, are uniform finite and continuous for
all points within S and on its boundary. Then, proceeding as in § 16 and
taking account of the conditions to which P and Q are subject, we have

/Yp\7-r)7 ,7 fvfiQj d$j\ [[fdpdQ , dP3Q\j j
PV-Qdxdy = P (~ dy - ~ dx }- U- —• + -^ ~ dacdy ;

Jj J \<Mt ' dy ) JJ\dx dx oydyj

dy ) JJ\dx dx oydy

* The element dn of the normal is, by this definition, measured inwards to, or outwards
from, the area S according as the particular boundary-line is described in the positive, or in the
negative, trigonometrical sense. Thus, if S be the space between two concentric circles, the
element dn at each circumference is drawn towards its centre ; the directions of integration are
as in § 2.

412 PRELIMINARY LEMMAS [216.

32 32
where V2 denotes = — h ^-- , the double integrals extend over the area of S, and

dx2 dy2

the single integral is taken round the whole boundary of S in the direction
that is positive for the bounded area S.

Let ds be an element PT of arc of the boundary at a point (x, y), and dn be
an element TQ of the normal at T drawn to the
interior of the space included by the boundary-
line regarded as a simple closed curve ; and let ty
be the inclination of the tangent at T. Then in
(i), as TQ is drawn to the interior of the area in- P p

eluded by the curve, the direction of integration lg- 78-

being indicated by the arrow (so that S lies within the curve), we have

dx = ds cos ^r — dn sin ty, dy = ds sin ty + dn cos i|r ;
and therefore it follows that, for any function R,

dR dR . dR

tr— = — TT— Sin W + -^— COS -vjr.

dn dx dy

Now for variations along the boundary we have dn = 0, so that

dR,_dRj 3E ,

~"~ ~^r CvO — /•* CvtJ ~~* *"— \JjvU,

on ox oy

And in (ii), as TQ is drawn to the interior of the area included by the curve,
the direction of integration being indicated by the arrow (so that 5 lies
without the curve), we have

dx = (— ds) cos -vjr + dn sin -fr, dy — (— ds) sin -v/r — dn cos ty,

dR dR . dR

and therefore -^— = =— sin y — ^— cos Y,

dn ox oy

so that, as before, for variations along the boundary,

dR . dR . dR ,
— ^- ds = ^— dy — -=- dx.
on ox oy

Hence, with the conventions as to the measurement of dn and ds, we have

both integrals being taken round the whole boundary of S in a direction that
is positive as regards S. Therefore

- - f P * * - \i(^ % + 3/
J on JJ\ox dx dy oy

In the same way, we obtain the equation

f[nv» vj i fn

QV2Pdxdy = - IQ
JJ J

a

dn JJ\dx dx dy dy

and therefore (PV2Q - QV'P) d^?y = ff Q |? - P ^) cfo,

y

216.] FOR SCHWARZ'S PROOF 413

where the double integral extends over the whole of 8, and the single
integral is taken round the whole boundary of S in the direction that is
positive for the bounded area S.

Now let u be a potential function defined as in the lemma; then u
satisfies all the conditions imposed on P, as well as the condition V2w = 0
throughout the area and on the boundary. Let Q = I ; so that V2Q — 0,

— = 0. Each element of the left-hand side is zero, and there is no dis-

dn

continuity in the values of P and Q ; the double integral therefore vanishes,
and we have

f 5- & =s 0,

J dn
the result Avhich was to be proved.

But if the derivatives of u are not required to satisfy the boundary
conditions, the foregoing equation may not be inferred ; we then have the
following proposition.

LEMMA III. Let u be a function, which is only required to satisfy the
general conditions for an area S ; and let u' be any other function, which
is required to satisfy the general conditions for that area and may or may
not be required to satisfy the boundary conditions. Let A be an area entirely
enclosed in S and such that no point of its whole boundary lies on any part of
the whole boundary of S ; then

|V duf ,du\, „

(u •= u 5- )ds = 0,

J\ dn on)

where the integral is taken round the whole boundary of A in a direction
which is positive with regard to the bounded area A, and the element dn of
the normal to a boundary-line is drawn towards the interior of the space
enclosed by that boundary-line, regarded merely as a simple closed curve.

The area A is one over which the functions u and u' satisfy the general
conditions. The derivatives of these functions satisfy the boundary-conditions
for A, because they are uniform, finite and continuous for all points inside S,
and the boundary of A is limited to lie entirely within S. Hence

the integrals respectively referring to the area of A and its boundary in a
direction positive as regards A. But, for every point of the area, V2w = 0,
W = 0 ; and u and u' are finite. Hence the double integral vanishes, and
therefore

(f

l(
J\

du' ,du

o -- u ^-
dn dn

taken round the whole boundary of A in the positive direction.

414 POTENTIAL FUNCTION [216.

One of the most effective modes of choosing a region A of the above
character is as follows. Let a simple curve (7j be drawn lying entirely within
the area S, so that it does not meet the boundary of 8; and let another
simple curve C2 be drawn lying entirely within Cl, so that it does not meet
(7j and that the space between C\ and (72 lies in S. This space is an area of
the character of A, and it is such that for all internal points, as well as for
all points on the whole of its boundary (which is constituted by C^ and (72),
the conditions of the preceding lemma apply. The curve (72 in the above
integration is described positively relative to the area which it includes : the
curve C-i is described, as in § 2, negatively relative to the area which it
includes. Hence, for such a space, the above equation is

// du' , du\ , (( du' , du\

MM 5 u 5- ] dSi - nu -= u 5- }ds2 = 0,

J\ dn Cn) J\ dn dnj

if the integrals be now extended round the two curves in a direction that is
positive relative to the area enclosed by each, and if in each case the normal
element dn be drawn from the curve towards the interior.

217. We now proceed to prove that a function u, required to satisfy the
general conditions for an area included within a circle, is uniquely determined
by the series of values assigned to u along the circumference of the circle.

Let the circle 8 be of radius R and centre the origin. Take an internal
point z0 = reP1, and its inverse z0' = r'e^ (such that rrf = R°) : so that z0' is
external to the circle. Then the curves determined by

— Zr,

for real values of X, are circles which do not meet one another. The boundary
of 8 is determined by X = 1, and X = 0 gives the point z0 as a limiting circle :
and the whole area of S is obtained by making the real parameter X
change continuously from 0 to 1.

Lemma III. may be applied. We choose, as the ring-space, the area
included between the two circles determined by Xj and X2, where

1 > Xj > X, > 0 ;
and then we have

[( du' ,du\ 1 (I du'
llu -^ -- u — \ds1 = l(u ^ -- u
J \ on dnj J \ dn

,du\
5-

on]

where the integrals are taken round the two circumferences in the trigono-
metrically positive direction (dn being in each case a normal element drawn
towards the centre of its own circle), and the function u' satisfies the general
and the boundary conditions for the ring-area considered. Moreover, the
area between the circles, determined by \ and X2, is one for which u satisfies

217.] DETERMINED FOR A CIRCLE 415

the general conditions, and its derivatives certainly satisfy the boundary
conditions : hence

fdu .. fdu

U- cfej = 0, 1 5- ds2 = 0.

J dn J dn

Now the function u' is at our disposal, subject to the general conditions
for the area between the two X-circles and the boundary conditions for each
of those circles. All these conditions are satisfied by taking u' as the real

(z — z \
— -, j , that is, in the present case,
Z ZQ /

, , I * - Ji
u =log z~-^, .

(T \
P Xj j , so

that

u' ^— dsl = 0 :

J dn

and similarly for all points on the inner circle u' is equal to the constant

/ r \

log I p X2 ) , so that
\zi /

/«'|>=o.

Again, for a point z on the outer circle, whose angular coordinate is ^r,

du'

the value of ^— for an inward drawn normal is (5 11)
dn

(E2-r2V)2

_
\,R (R2 - r2) {R2 - 2Rr\ cos (^ - 0) + r2^2} '

and because the radius of that outer circle is \R (R- — r2)/(R2 — r2\12), we
have

, \1R(R2-r2),

Denoting by/(X!, x|r) the value of u at this point i/r on the circle determined
by Xj, we have

(- « + ^v

Similarly for the inner circle, the normal element again being drawn towards
its centre, we have

Combining these results, we have

2 — ^ 2

T27
Jo

416 INTEGRAL-EXPRESSION [217.

In the analysis which has established this equation, Xj and X2 can have all
values between 1 and 0 : the limiting value 0 is excluded because then u'
is not finite, and the limiting value 1 is excluded because no supposition has
been made as to the character of the derivatives of u at the circumference
of 5.

The equation which has been obtained involves only the values of u
but not the values of its derivatives. Since the values of u are finite both
for X = 0 and \ = 1, and the integrals are finite, the exclusion of the limiting
values of X need not be applied to the equation, although the exclusion was
necessary during the proof, owing to the presence of quantities that have
since disappeared. Hence the equation is valid when we take \ = 1, X2 = 0.

When X*, — 0, the corresponding circle collapses to the point z0 : the value
of y(X2, ty) is then the value of u at #0 say u(r, <£); and the integral
connected with the second circle is ZTTU (r, <£).

When Xj = 1, the corresponding circle is the circle of radius R ; the value
of /(Xj, ty) is then the assigned value of u at the point i/r on the circum
ference, say the function /(^). Substituting these values, we have

u ^ V=L

the integral being taken positively round the circumference of the circle 8.

It therefore appears that the function u, subjected to the general
conditions for the area of the circle, is uniquely determined by the values
assigned to it along the circumference of the circle.

The general conditions for u imply certain restrictions on the boundary
values. These values must be finite, continuous and uniform : arid therefore
as a function of ty, must be finite, continuous, uniform and periodic in
of period 2-7T.

218. It is easy to verify that, when the boundary values f(ty) are not
otherwise restricted, all the conditions attaching to u are satisfied by the
function which the integral represents.

Since the real part of (Re^ + z)j(Re^ - z) is the fraction

(R> - r2)/{E2 - 2Rr cos (^ - <£) + r2},
it follows that u is the real part of the function F (z), defined by the equation

„. .

F(z\= ^-77
^ ' l

For all values of z such that z < R, the fraction can be expanded in a series
of positive integral powers of z, which converges unconditionally and uni
formly ; and therefore F (z) is a uniform, continuous, analytical function,

218.] FOR A POTENTIAL FUNCTION 417

everywhere finite for such values of z. Hence all its derivatives are uniform,
continuous, analytical functions, finite for those values of z\ and these
properties are possessed by the real and the imaginary parts of such

dm+n U (frn+n p /z\

derivatives. Now ^-^ is the real part of ^.-j—L-Zj and therefore,

for all integers m and n positive or zero, it is a uniform, finite and continuous
function for points such that z\<R, that is, for points within the circle.
Moreover, since u is the real part of a function of z, and has its differential
coefficients uniform, finite and continuous, it satisfies the differential equation
V"u = 0.

To infer the continuity of approach of u(r, <£) to /(<£) as r is made equal
to R, we change the integral expression for u (r, <£) into

Moreover for all values of r < R (but not for r = R), we have

1 /•2"--<£ ft2 — r2 If (7?4-r

^ P^?- ~l^«de= ^n- H±Ttani =1;

%TT J -$ R2 — 2Rr cos V -f r" TT [_ (R — r

and therefore

/-«(«•.+)-/<+)

7?2

Let ® denote the subject of integration in the last integral. Then, as r
is made to approach indefinitely near to R in value, © becomes infinitesimal
for all values of 6 except those which are extremely small, say for values of 0
between - $ and + 8. Dividing the integral into the corresponding parts,
we have

Let M be the greatest value of /(i/r) for points along the circle. Then the
first integral and the second integral are less than

_ j

27T £-a

respectively ; by taking r indefinitely near to R in value, these quantities
can be made as small as we please. For the third integral, let k be the
greatest value of f(<f> + 6) -/(</>) for values of 6 between 8 and - 8 : then the
third integral is less than

k fs R?-r*

2-7T J _5 R*-* '

that is, it is less than ±- tan"1 (^ -^~ §j • so that, when r is made nearly
equal to R, the third integral is less than k.

F' 27

418 DISCONTINUITY IN VALUE [218.

If then k be infinitesimal, as is the case when /((/>) is everywhere finite
and continuous, the quantity / can be diminished indefinitely; hence u(r, (/>)
continuously changes into the function /(<£) as r is made equal to R. The
verification that the function, defined by the integral, does satisfy the general
conditions for the area of the circle and assumes the assigned values along
the circumference is thus complete.

Ex. Shew that, if M denote the maximum value (supposed positive) of/(^) for points
along the circumference of the circle and if u (0) denote the value of the function at the
centre, then

I u (r, d>) - u (0) | < - M sin-1

also that, if u (0) vanish, then

u (r, $)< - Mtan~l ^ . (Schwarz.)

219. But in view of subsequent investigations, it is important to consider
the function represented by the integral when the periodic function /(</>)
which occurs therein is not continuous, though still finite, for all points on
the circumference. The contemplated modification in the continuity is that
which is caused by a sudden change in value of /(<£) as <j> passes through a]
value a : we shall have

/(« + •)-/(«-•)- -A,
when e is ultimately zero. Then the following proposition holds :

Let a function f($) be periodic in ZTT, finite everywhere along the circle, j
and continuous save at an assigned point a where it undergoes a sudden increase
in value : a function u can be obtained, which satisfies the general conditions
for the circle except at such a point of discontinuity in the value of f((f>\ and
acquires the values of f((f>) along the circumference.

Let p be a quantity < R : then along the circumference of a circle of radius
p, the general conditions are everywhere satisfied for the function u, so that, if
u (p, i/r) be the value at any point of its circumference, the value of u at any
internal point is given by

u (r, $} = ~ £ u (p, W p2_2/5rcPos"(^_</)) + r2 <**'

Now p can be gradually increased towards R, because the general conditions
are satisfied; but, when p is actually equal to R, the continuity of
u(p, ^r) is affected at the point a. We therefore divide the integral into
three parts, viz., 0 to a - e, a - e to a + e, and a + e to 2-n-, when p is very
nearly equal to R. For the first and the third of these parts, p can, as in the
preceding investigation, be changed continuously into R without affecting
the value of the integral. If we denote by p the integral

~

where the range of integration does not include the part from a — e to a + e,

219.] ALONG THE CIRCUMFERENCE 419

and where the values /(a — e), /(a + e) are assigned to u (R, a — e), u' (R, a. + e),
respectively ; the sum of the integrals for the first and the third intervals is
p + A, where A is a quantity that vanishes with R — p, because the subject of
integration is everywhere finite. For the second interval, the integral is
equal to q + A', where

1 [a+f R2 — r2

and A' is a quantity vanishing with R — p because the subject of integration
is everywhere finite. So far as concerns q, let M be the greatest value of

l/Wh ^en

, MR+r

l?l<2ir.B-r '

a quantity which, because M is finite (but only if M be finite), can be made
infinitesimal with e, provided r is never actually equal to R. If then, an
infinitesimal arc from a — e to a + e be drawn so as, except at its assigned
extremities, to lie within the area of the circle, the last proviso is satisfied :
and the effect is practically to exclude the point a from the region of
variation of u as a point for which the function is not precisely defined.
With this convention, we therefore have

2?r

_ ~

so that, by making p ultimately equal to R and e as small as we please, the
difference between u (r, <£) and the integral defined as above can be made zero.
Hence the integral is, as before, equal to the function u (r, 0), provided that
the point a be excluded from the range of integration, the value /(a— e) just
before ^=0. and the value /(a+e) just after ^=0. being assigned to u'(R, ^r).
It therefore appears that discontinuities may occur in the boundary
values when the change is a finite change at a point, provided that all
the values assigned to the boundary function be finite.

COROLLARY. The boundary value may have any limited number of points
of discontinuity, provided that no value of the function be infinite and that at
all points other than those of discontinuity the periodic function be uniform,
finite and continuous : and the integral will then represent a potential function
satisfying the general conditions.

The above analysis indicates why discontinuities, in the form of infinite
values at the boundary, must be excluded: for, in the vicinity of such a
point, the quantity M can have an infinite value and the corresponding
integral does not then necessarily vanish. Hence, for example, the real
part of

is not a function that, under the assigned conditions, can be made a boundary
value for the function u.

27—2

420 SPECIAL FORMS [219.

It is easy to construct a function with permissible discontinuities. We know (§ 3)
that the argument of a point experiences a sudden change by p

TT when the path of the point passes through the origin. Let
a point P on a circle be considered relative to A : the inclina- Q /
tion of AP to the normal, drawn inwards at A, is - - £(o - <£),

and of A Q to the same line is - | - |(a - 0') , so that there

is a sudden change by TT in that inclination. Now, taking a function

A^ ,r fir 'i/ ,01
# W>)= tan-1 tan \=- f (a- <p}\ I,

7T L. {^ J J

and limiting the angle, defined by the inverse function, so that it lies between - \n
and +^TT, as may be done in the above case and as is justifiable with an argument
determined inversely by its tangent, the function g (0) undergoes a sudden change A as <p
increases through the value a. Moreover, all the values of g (<£) are finite : hence g (<£) is
a function which can be made a boundary value for the function u. Let the function
thence determined be denoted by ua.

By means of the functions ua, we can express the value of a function u whose boundary
value /(<£) has a limited number of permissible discontinuities. Let the increases in value
be Alt ...,Am at the points al5 a2, ... , aTO respectively : then, if gn(<j>) denote

we have gn (an + f}-gn («« - * ) = An, when e is infinitesimal. Hence

has no discontinuity at an, that is,/(0)-^(0) has no discontinuity at an.

Hence also /(<£)- 2 #„(<£) has no discontinuity at a1? ..., am, and therefore it is

n~ 1

uniform, finite, and continuous everywhere along the circle; and it is periodic in 27r.
By § 218, it determines a function U which satisfies the general conditions.

Each of the functions gn (0) determines a function un satisfying the general conditions :
hence, as u is determined by /(<£), we have

m

U- S Un=U,
n=l

which gives an expression for u in terms of the simpler functions un and of a function U
determined by simpler conditions as in § 218.

Ex. Shew that, if /(\|^) = 1 from - \ir to +\n and =0 from +^TT to f TT, then « is the
real part of the function

1. 1+12

— log -.-

^7^ "

The general inference from the investigation therefore is, that a function
of two real variables x and y is uniquely determined for all points within a
circle by the following conditions :

(i) at all points within the circle, the function u and its derivatives

du du &u &u be unifo finite anc} continuous, and

dx' dy' dx*' 8y2

must satisfy the equation V2i* = 0 :

(ii) if /((£) denote a function, which is periodic in </> of period 2?r, is
finite everywhere as the point <f> moves along the circumference,

219.] GENERAL PROPERTIES 421

is continuous and uniform at all except a limited number of
isolated points on the circle, and at those excepted points
undergoes a sudden prescribed (finite) change of value, then
to u is assigned the value /(<£) at all points on the circumference
except at the limited number of points of discontinuity of that
boundary function.

And an analytical expression has been obtained, the function represented by

which has been verified to satisfy the above conditions.

220. We now proceed to obtain some important results relating to a
function u, defined by the preceding conditions.

I. The value of u at the centre of the circle is the arithmetic mean of its
values along the circumference.
For, by taking r = 0, we have

the right-hand side being the arithmetic mean along the circumference.

II. If the function be a uniform constant along the circumference, it is
equal to that constant everywhere in the interior.

For, let C denote the uniform constant ; then

= 0
for all values of r less than R, that is, everywhere in the interior.

But if the function, though not varying continuously along the circum
ference, should have different constant values in different finite parts, as, for
instance, in the example in § 219, then the inference can no longer be drawn.

III. If the function be uniform, finite and continuous everywhere in the
plane, it is a constant.

Since the function is everywhere uniform, finite and continuous, the
radius R of the circle of definition can be made infinitely large : then, as
the limit of the fraction (R2 — r-)j{R2 — 2Rr cos (ty — (£) + r2} is unity, we

have

1 f2"
u(r, <£)=2^.J w(co,xjr)cty,

the integral being taken round a circle of infinite radius whose centre is the
origin. But, by (I) above, the right-hand integral is u (0), the value at the
centre of the circle ; so that

u.(r, <j>) = u(Q),

and therefore u has the same value everywhere.

This is practically a verification of the proposition in § 40, that a uniform,
finite and continuous function w, which has no infinity anywhere, is a constant.

422 GENERAL PROPERTIES [220.

IV. A uniform, finite and continuous function u cannot have a maximum
value or a minimum value at any point in the interior of a region over which,,
subject to the general conditions as to the differential coefficients, it satisfies the
differential equation V2M = 0.

If there be any such point not on the boundary, it can be surrounded by
an infinitesimal circle for the interior of which, as well as for the circum
ference of which, u satisfies both the general and the boundary conditions ; hence

[du 7 A
I ~- ds = 0,

J on

the integral being taken round the circumference. But in the immediate

p

vicinity of such a point, ~- has everywhere the same sign, so that the
integral cannot vanish : hence there is no such point in the interior.

In the same way, it may be proved that there cannot be a line of
maximum value or a line of minimum value within the surface : and that
there cannot be an area of maximum value or an area of minimum value
within the surface.

V. It therefore follows that the maximum values for any region are to be
found on its boundary : and so also are the minimum values.

If M be the maximum value, and if m be the minimum value of the
function for points along the boundary, then the value of the function for an
interior point is < M and is > m and can therefore be represented in the form
Mp + m (1 — p), where p is a real positive proper fraction, varying from point
to point.

In particular, let a function have the value zero for a part of the
boundary and have the value unity for the rest : the value that it has for
points along a line in the interior is always positive and has an upper limit
q, a proper fraction. But q will vary from one line to another. If the region
be a circle and q be the proper fraction for a line in the circle, then the value
along that line of a function u, which is still zero over the former part of the
boundary but has a varying positive value ^. p along the remainder, is
evidently ^ qp. This fraction q may be called the fractional factor for the
line in the supposed distribution of boundary values.

VI. It may be noted that the second of these propositions can now
be deduced for any simply connected surface. For when a function is
constant along the boundary, its maximum value and its minimum value
are the same, say X: then its value at any point in the interior is
\p + \(l — p), that is, X, the same as at -the boundary. Consequently if
two functions % and u2 satisfy the general conditions over any region, and
if they have the same value at all points along the boundary, then they
are the same for all points of the region. For their difference satisfies

220.] EXISTENCE FOR ANALYTICAL CURVE 423

the general conditions : it is zero everywhere along the boundary : hence
it is zero over the whole of the bounded region.

If, then, a function u satisfy the general conditions for any region, it is
unique for assigned boundary values that are everywhere finite, uniform, and
continuous except at isolated points.

221. The explicit expression of u with boundary values, that are
arbitrary within the assigned limits, has been determined for the area
enclosed by a circle : the determination being partially dependent upon the
form assumed in § 217 for the subsidiary function u'. The assumption of
other forms for u', leading to other curves dependent upon a parametric
constant, would lead by a similar process to the determination of u for the
area limited by such families of curves.

But without entering into the details of such alternative forms for u', we
can determine the value of u, under corresponding conditions, for curves
derivable from the circle by the principle of conformal representation*.
Suppose that, by means of a relation

or, say x + iy = p (f , 77) + iq (£, 77),

where p and q are real functions of £ and 77, the area contained within the
circle is transformed, point by point, into the area contained within another
curve which is the transformation of the circle : then the function u (x, y)
becomes, after substitution for x and y in terms of £ and 77, a function, say U,
of £ and 77.

Owing to the character of the geometrical transformation, p and q (and
their derivatives with regard to £ and 77) are uniform, finite and continuous
within corresponding areas. Hence

8 U _ du dp dudq 8 U _ du dp du dq _
8£ dx dj; dy 8^ ' 8/7 dx 877 dy 877 '

d*U dn-U (&u 8%\ (78/A2 /8»V

and aS" + -5-5 = a"* + 51 1 U6- + a

dp 8772 \dx2 dy2J \\di-J \drjj

so that the function U satisfies the general conditions for the new area
bounded by the new curve.

Moreover, u has assigned values along the circular boundary which is
transformed, point by point, into the new boundary ; hence U has those
assigned values at the corresponding points along the new boundary. Thus
the function U is uniquely determined for the new area by conditions which
are exactly similar to those that determine u for a circle : and therefore the

* The general idea of the principle, and some illustrations of it, as expounded in
Chapters XIX and XX, will be assumed known in the argument which follows : see especially
§§ 265, 266.

424 POTENTIAL FUNCTION [221.

potential function is uniquely determined for any area, which can be con-
formally represented on the area of a circle, by the general conditions of
§ 216 and the assignment of values that are finite and, except at a limited
number of isolated points where they may suffer sudden (finite) changes of
value, uniform and continuous at all points along the boundary of the area.

One or two examples of very special cases are given, merely by way of
illustration. The general theory of the transformation of a circle or an
infinite straight line into an analytical curve will be considered in Chapter
XX. But, meanwhile, it is sufficient to indicate that, by the principle of
conformal representation, we can pass from the circle to more general curves
as the boundary of an area within which the potential function is defined by
conditions similar to those for a circle : in particular that, by assuming the
result of §§ 265, 266, we can pass from the circle to an analytical curve as the
boundary of such an area.

Ex. 1. A function u satisfying the general conditions for a circle of radius unity and
centre the origin, and having assigned values /(^r) along the circumference, is determined at
any internal point by the equation

"

Now the circle and its interior are transformed by the equation

•+1-3
r

into a parabola and the excluded area (Ex. 7, § 257) : so that, if R, 6 be polar coordinates
of any point in that excluded area, we have

Corresponding to the circle /• = !, we have the parabola

Rca&\6 = \;
if 0 determine the point on the parabola, which corresponds to \|/- on the circle, we have

or TJr = Q.

Hence the function U(R, 6) assumes the values /(9) along the boundary of the
parabola.

Also l-r2 = ^(R*cos$d-l),

1 - 2r cos ty - 0) + r2 =-i [R cos2 £0 - 2/2* cos £0 cos £(e + 6) + 1] ;

Mm

and therefore we have the following result :

A function ivhich satisfies the general conditions for the area bounded by and lying on the
convex side of the parabola Rcos2^Q = l and is required to assume the value /(6) at points
along the parabola, is defined uniquely for a point (r, 6} external to the parabola by the
integral

The function /(O) may suffer finite discontinuities in value at isolated points: elsewhere
it must be finite, continuous and uniform.

221.] FOR CONFORMALLY RELATED AREAS 425

Ex. 2. Obtain an expression for u at points within the area of the same parabola, by
using

as the equation of transformation of areas (§ 257).
Ex. 3. When the equation

is used, then, if z = x+iy and £=X+iY, we have

. _

+ iy~

If the point £ describe the whole length of the axis of X from - oo to + oo , so that we
may take f=JT=tan^> with $ increasing from -\K to +%TT, we have #=cos2</>,
y = sin20; and z describes the whole circumference of a circle, centre the origin and
radius unity, in a trigonometrically positive direction beginning at the point ( - 1, 0). We
easily find

rcos0 rsind r2 1

where £ = RcosQ, r) = RsmQ. Moreover, for variations along the circumference, we
have i|r = 2<£; whence, substituting and denoting by F(x), =/(2 tanr1;?), the value of
the potential at a point on the axis of real quantities whose abscissa is x, we ultimately
find

as the value of the potential-function u at a point (R, Q) in the upper half of the plane,
when it has assigned values F(x] at points along the axis of real variables.

222. The function u has now been determined, by means of the general
conditions within an area and the assigned boundary values, for each space
obtained by the method indicated in § 221. But the determination is
unique and distinct for each space thus derived ; and, if two such spaces
have a common part, there are distinct functions u. We now proceed to
shew that when two spaces, for each of which alone a function u can be
determined, have a common part which is not merely a point or a line,
then the function u is uniquely determined for the combined area by the
assignment of finite, uniform and continuous values {or partially discontinuous
values, as in § 219) along the boundary of the combined area.

Let the spaces be 2\ and jP2 having a
common part T, so that the whole space
can be taken in the form Tl + T2 - T. Let
the part of the boundary of T^ without T^
be L0, and the part within T2 be L.2 : and
similarly, for the boundary of T2, let L^ de
note the part within Ta and L3 the part
without it. Then the boundary of

I T1+T>-T Fig-80'

is made up of L0 and L3 : the boundary of T is made up of L: and L,.

426 COMBINATION [222.

With an assignment of zero value along LQ and unit value along L.2,
let the fractional factor (§ 220, V), for the line L± in the region T^ be q± ;
and with an assignment of zero value along Ls and unit value along Llt let
the fractional factor along the line L2 in the region Tz be q2. Then q1 and qz
are positive proper fractions.

Let any series of values be assigned along L0 and L3 subject to the
conditions of being uniform, finite everywhere, and discontinuous, if at all,
only at a limited number of isolated points ; these values are the boundary
values of the function u to be determined for the whole area, and will be
called the assigned values. Let the maximum of the values be M and the
minimum be m ; and denote M — m by /*, so that p is positive.

Assume, for a boundary value along L2, the minimum m of the assigned
values for the function along L0 and L3. Let the function, which is uniquely
determined for the region 2\ by the general conditions for the area and by
values along the boundary, constituted by the assigned values along L0 and
the assumed value m along L2, be denoted by i^. The values assumed by u^
along the line L± in this region are uniform, finite and continuous ; and they
may be denoted by m+p/j,, where p is a positive proper fraction varying from
point to point along the line.

Let the function, which is uniquely determined for the region T2 by the
general conditions for the area and by values along the boundary, constituted
by the assigned values along L3 and by the values of Wj along L1} be denoted
by uz. Then the uniform, finite, continuous values which it assumes along
L2 are of the form m + qp, where q is a positive proper fraction varying from
point to point along the line ; let the greatest of these values be m + Q/J,,
where Q lies between 0 and 1.

For the region T± determine a function* u3 by means of boundary values,
consisting of the assigned values along L0 and the values of u2, viz., m + Q/J,,
along -Z/j. Then the function u3 — u-^ satisfies the general conditions ; its
value along the part L0 of the boundary is zero, and its value along the
other part L2 of the boundary is < Q/JL and is greater than zero. Hence u3— u±
is always positive within T± , and along L^ we have u3 — u^ ^ qiQ/J*.

For the region T2 determine a function M4 by means of boundary values,
consisting of the assigned values along L3 and the values of u3 along L^
Then the function u4 — u2 satisfies the general conditions; its value is zero
along Z3; and its value along L^ is that of u3 — ult that is, a positive quantity
which is not greater than qfyfi. Hence w4 — u2 is always positive within T2,
and along L.2 we have u4 - u2

* All the succeeding functions will be determined subject to the general conditions for
the respective areas ; the specific mention of the general conditions will be omitted.

222.] OF AREAS 427

For the region T^ determine a function u5 by means of boundary values,
consisting of the assigned values along L0 and the values of w4 along L2.
Then the function u5 — us satisfies the general conditions ; its value is zero
along L0\ and its value along L2 is that of w4 — w2, that is, a positive quantity
which is not greater than q2qiQp. Hence u5 — u3 is always positive within T^ ,
and along L1 we have us — u3 ^ (ffli'QfL

Proceeding in this manner for the regions alternately, we obtain functions
UMI+I for the region T1} such that um+l has the assigned values along L0 and
the values of u.M along L2 ; and functions u,2n for the region T2, such that u^n
has the assigned values along L3 and the values of um-i along L^. And the
functions are such that

Man-i > 0 in ^ and ^ q^q^1 Qp along L± ; and
— u-2n > 0 in T2 and ^ q^q.™ Qp along L2.

Hence, both for functions with an uneven suffix and for functions with an
even suffix, there are limits to which the functions approach along L± and L.2
respectively ; let these limits be u' and u".

Both of these limits are finite ; for along L1} we have

u' = HI + (u3 - MJ) + (u, — u3) + . . . ad inf.

so that this expression, which is finite, is an upper limit and m is a lower
limit for u'. And, along L2, we have

u" = u2 + (w4 — M2) + (u6 — w4) -f ... ad inf.
<; m + Q/J, + qiq2Q/j, + q^q22 Q/J, + ...

^ml Qn

so that this expression, which is finite, is an upper limit and m is a lower
limit for u". Hence both u' and u" are finite.

Now in determining u' for Tl and regarding it as the limit of u2n+1, we
have its values along L2 as the values of w2n, that is, of u" in the limit ; and
in determining u" for T2 and regarding it as the limit of u2n+2, we have its
values along D, as the values of u2n+1, that is, of u in the limit. Hence over
the whole boundary of T, the region common to Tl and Ta, we have u' = u" ;
and therefore (by § 220, VI) we have u =u" over the whole area of the
common region T.

Lastly, let a function u be determined for the region T1} having the
assigned values along L0 and the values of u' along L.2. Then the function
u - u' satisfies the general conditions ; it has zero values round the whole

428 COMBINATION OF AREAS [222.

boundary of Tl, and therefore (by § 220, VI) it is zero over the whole region
T!. Hence u' is the function for Tl.

Similarly, determining a function u for T2, having the assigned values
along Ls and the values of u" along Llf we have u = u" everywhere in T2, so
that u" is the function for T^.

The functions u' and u" satisfy the general conditions for Tl and T2
respectively ; and these two regions have a common portion T over which
•u! and u" have been proved to be the same. Hence, by Lemma I. of § 216,
they determine one and the same function for the whole region combined of
Tl and T2 ; this function u satisfies the general conditions and, along the
boundary of the whole region, assumes values that are assigned arbitrarily
subject only to the general limitations of being everywhere finite and,
except for finite discontinuities at isolated points, uniform and continuous.
The proposition is therefore established.

This method of combination, dependent upon the alternating process
whereby a function determined separately for two given regions having a
common part is determined for the combination of the regions, is capable of
repeated application. Hence it follows that a function exists, subject to the
general conditions within a given region and acquiring assigned finite values
along the boundary of the region, when the region can be obtained by
combinations of areas that can be conformally represented upon the area of a
circle.

Note. Let A, B, C be three non- intersecting simple closed curves, such
that G lies within B and B within A. The area bounded by the curves A and
C can, by a similar method, be combined with the whole area enclosed by B ;
and we can make the same inference as above, as to the existence of a function
u for the whole area enclosed by A, when it exists for the areas that are
combined.

223. At the beginning of the discussion it was assumed that the areas,
in which the existence of the function is to be proved, lie in a single sheet
(§ 216) or, in other words, that no branch-point occurs within the area.

It is now necessary to take the alternative possibility into consideration :
a simple example will shew that the theorem just proved is valid for an area
containing a branch-point except in one unessential particular.

Let the area be a winding surface consisting of m sheets : the region in
each sheet will be taken circular in form, and the centre c of the circles will
be the winding-point, of order m— I. Such a surface is simply connected
(§ 178) ; and its boundary consists of the m successive circumferences which,
owing to the connection, form a single simple closed curve. Using the
substitution

z - c = RZM,

223.] BRANCH-POINT IN AREA 429

we have a new ^-surface which consists of a circle, centre the Z-origin and
radius unity : it lies in one sheet in the ^-region and has no branch-points ;
its circumference is described once for a single description of the complete
boundary of the winding-surface. The correspondence between the two
regions is point-to-point: and therefore the assigned values along the bound
ary of the winding-surface lead to assigned values along the ^-circumference.
Any function w of z changes into a function W of Z: hence u changes
into a real function U satisfying the general conditions in the ^-region ;
and conversely.

But a function U, satisfying the general conditions over the area of a
plane circle and acquiring assigned finite values along the circumference, is
uniquely determinate ; and hence the function u is uniquely determined on
the circular winding-surface by satisfying the general conditions over the area
and by assuming assigned values along its boundary.

It is thus obvious that the multiplicity of sheets, connected through
branch-lines terminated at branch-points and (where necessary) on the single
boundary of the surface consisting of the sheets, does not affect the validity
of the result obtained earlier for the simpler one-sheeted area ; and therefore
the function u, acquiring assigned values along the boundary of the simply
connected surface and satisfying the general conditions throughout the area
of the surface which may consist of more than a single sheet is uniquely deter-
\ minate.

There is, as already remarked, one unessential particular in which
deviation from the theorem occurs when the region contains a branch-point.
At a branch-point a function may be finite*, but all its derivatives are not
necessarily finite ; and therefore at such a point a possible exception to the
general conditions arises as to the finiteness of value of the derivatives
and the consequent satisfying of the equation V-u = 0 : no exception, of
course, arises as regards the uniformity of the derivatives on the Riemann's
surface. The exception does not necessarily occur ; but, when it does occur,
it is only at isolated points, and its nature does not interfere with the validity
of the proposition. We shall therefore assume that, in speaking of the
general conditions through the area, the exception (if necessary) from the
general conditions, of finiteness of value of the derivatives at a branch-point, is
tacitly implied.

Hence we infer, by taking combinations of circles in a manner some
what similar to the process adopted for successive circles of convergence
in the continuation of a function in § 34, that a function u exists, subject to the
general conditions within any simply connected surface and acquiring assigned
finite values along the boundary of the surface.

* Infinities of the function itself at a branch-point will fall under the general head of infinities
of the function, diecnssed afterwards (in § 229).

430

MODULI OF PERIODICITY

[224.

224. The functions which have been discussed so far in the present
connection are functions which have no infinities and, except possibly at
points on the boundaries of the regions considered, no discontinuities : they
are uniform functions. And the regions have, hitherto, been supposed simply
connected parts of a Riemann's surface, or simply connected surfaces. When
the surface is multiply connected, we resolve it by a canonical system
(§ 181) of cross-cuts and proceed as follows.

We now proceed to introduce the cross-cut constants, and so to consider
the existence of functions which have the multiform character of the integrals
of uniform functions of position on the Riemann's surface. The functions
will still be considered to be uniform, finite and continuous except at the
cross-cuts : their derivatives will be supposed uniform, finite, and continuous
everywhere in the region, and subject to the equation V2u = 0 : and boundary
values will be assigned of the same character as in the previous cases. As
moduli of periodicity are to be introduced, the unresolved surface is no longer
one of simple connection : we shall begin with a doubly connected surface.

Let such a surface T be resolved, in two different ways, into a simply
connected surface : say into Tt by a cross-cut Qx , and into rl\ by a cross
cut Q2. Mark on T^ and on T2 the directions of Q2 and of Q1 respectively : the

Fig. 81.

notations of the boundaries are indicated in the figures, and T' is the
region between the lines of Qj and Q2.

It will be shewn that a function u exists, determined uniquely by the
following conditions :

(i) The first and the second derivatives are throughout T to be
uniform, finite and continuous, and to satisfy V"u = 0 : but no conditions
for them are assigned at points on the boundary :

(ii) The (single) modulus of periodicity is to be K, which will be
taken as an arbitrary, real, positive constant : the value of any branch of u at
a point on the positive edge is therefore to be greater by K than its value at
the opposite point on the negative edge :

(iii) Some selected branch of u is to assume assigned values along

224.] FOR MULTIPLY CONNECTED SURFACES 431

a and b', typically represented by H, and assigned values along a and b,
typically represented by G. These boundary values are to be finite every
where, though they may be discontinuous at a finite number of isolated points
on the boundary ; such discontinuity will arise through the modulus.

In T1} for zero values along a, b, a', b' and for unit values along Q{~
and Qj+, let the fractional factor for the line Q.2 be q1 : and similarly in T2,
for zero values along a, b, a', b' and for unit values along Q2~ and Q2+,
let the fractional factor for the line Qi be q2, where <ft and g2 are positive
proper fractions.

For the simply connected region* T-i determine a function u1} satisfying the
general conditions and having as its boundary values, H along a' and b', G
along a and b, arbitrarily assumed values represented by 6 (the maximum
value being Mj. and the minimum value being m^ along Q~ and values
B + K along Qi+ : the function so obtained is unique. Let the values
along the line Q2 in ^i be denoted by w/.

For the region T2 determine a function w2, satisfying the general
conditions and having as its boundary values, H along a' and b', G — K
along a and 6, u^ — K along Q2~ and u-[ along Q2+ : the function so ob
tained is unique. Let its values along the line Ql in Ta be denoted by
u2', the maximum value being Mt and the minimum value being m2.

For the region Tl determine a function us , satisfying the general conditions
and having as its boundary values, H along a' and b', G along a and b, u2'
along Qr and u.2' + K along Qj+ : the function so obtained is unique. Let its
values along the line Q2 in Tt be denoted by u3'. Then the function us — u:
satisfies the general conditions in Tl ; it is zero along a' and b', a and b : it is
u,f - 0 along Qj- and also along &+, and w/ - 6 ^ M2 - m1 and ^m2-M^

225. A difference of limits for u3 — w/ arises according to the relative
values of M2 and ml, of ra2 and Mt ; evidently M.2 - ml > m2 - M1.

(i) If m2 — M1 be positive, then Mz — m-^ is positive and equal, say, to
X; the boundary values for w3 — Wj may range from 0 to X and we have
u3' - iii > 0 < q{\, along Q2.

(ii) If m2 — Ml be negative and equal to — e, then M2 — m^ is either
positive or negative.

(a) If M^ — m^ be negative, then the boundary values for u^ — ii^
may range from 0 to — e, that is, boundary values for Wj — u3 may range from 0 to
e and we have w/ — u3 > 0 < ^e along Q.2> which may be expressed in the form

\u3 -U1'\<q1e,
where e is the greatest modulus of values along the boundary.

* In the special case, when Tl is bounded by concentric circles and the cross-cut is made along
a diameter, the region can be represented conformally on the area of a circle : see a paper by the
author, Quart. Journ. Math., Vol. xxvi, (1892), pp. 145—148.

432 EXISTENCE OF POTENTIAL FUNCTION [225.

(b) If Ms - m1 be positive, let its value be denoted by 77 : then the
boundary values for u3 — u^ may range from 77 to — e. The boundary values for
uz — w.j + e may range from 0 to 77 + e, and it is a function satisfying all the
internal conditions : hence w3 — w: + e ^ ql (77 + e), and therefore

«s - MI ^ qtf - (1 - ft) e < ftiy.

Again, the boundary values of MJ — u3 + 77 may range from 77 + e to 0, and it is
a function satisfying all the internal conditions : hence MJ — u2 + 77 ^ ft (77 + e),
and therefore

MI - % < fte - (1 - qi) i] ^ fte-

Hence at points where u3 > u-^, so that u3 - u± is positive, we have u3 — u± < ft 77 ;
and at points where us <ult so that u± — u3 is positive, we have u^ — u3 ^ fte.

Every case can be included in the following result* : If p be the greatest
modulus of the values of u2' — 0 along the two edges of Q1 in T1} then

along Q.2, so that q^p, is certainly the greatest modulus of ua' — u^ along Q2.

226. For the region jP2 determine a function u4, satisfying the general
conditions and having as its boundary values, H along a' and b', Q — K along
a and 6, w3' — K along Q.2~ and u3f along Q.2+ : the function so obtained is
unique. Let its values along the line Qx be denoted by w/. Then the
function u4 — «2 satisfies the general conditions in T2 : it is zero along a and b',
a and b : it is uz' — w/ along Q2~ and also along Q2+, and along Q2 we have

Hence, after the preceding explanations, we have along Qj in To

M/— ?V|< MiA4-

Proceeding in this way for the regions alternately, we have for Tl a function
w2w+1, the boundary values of which are, H along a' and &', (7 along a and 6,
M^' along Q{~ and w^' + .K" along Q^ : and along Q2

<ft ft /*;

and for T2, a function M2n+2, the boundary values of which are, H along a' and
6', G — K along a and b, um+1' - K along Q.,~ and M2n+1' along Q.2+ : and
along Qj

Thus both the function u*n+l along Q2 and the function u.2n along Qi
approach limiting values ; let them be u' and u" respectively.

These limiting values are finite. For

Uzn+i = U-i + (U3 — M4) + (lls — M3) -f . . . + (U2n+1 ~ Man-i) 5

* Another method of proceeding, different from the method in the text, depends upon the
introduction of another fractional factor for Q», having the same relation to minimum values
as q1 to maximum values ; but it is more cumbersome, as it requires the continuous consideration
of the separate cases indicated.

226.] FOR MULTIPLY CONNECTED SURFACES 433

in the limit, when n is infinitely large, the sum of the moduli of the terms of
the series at points along Q2

1-flfc

so that the series converges and the limit of UM+I, viz. u', is finite. Similarly
for u".

Now consider the functions in the portions T— T' and T' of the
region T.

For T-T' we have um, (that is, u" in the limit), with values H
along a' and b', u' along Q + : and also w2W+1, (that is, u' in the limit),
with values H along a and b' and u" along Q~: thus u' and M" have
the same values over the whole boundary of T — T' and, therefore, through
out that portion we have u = u".

For T' we have u^, (that is, u" in the limit), with values G-K along
a and b and w'- K along Qa~ : and also u2n+1, (that is, M' in the limit), with
values G along a arid & and u" -f K along Q+. Thus over the whole boundary
of T' we have u' — u"=K: and therefore within the portion T' we have

Lastly, for the whole region T we take u = u. In the portion T-T1 we
have u = u' = u", and in the portion T' we have u = u' = u" + K; that is, the
function is such that in the region T^ the value changes from u" at Q{~ to u" + K
at Qi+, or the modulus of periodicity is K.

Hence the function is uniquely determined for a doubly connected surface
by the general conditions, by the assigned boundary values and by the
arbitrarily assumed real modulus of periodicity.

227. We now consider the determination of the function, when the
surface S is triply connected and has a single boundary.

Let S be resolved, in two different ways, into a doubly connected surface.
Let Q1 be a cross-cut, which changes the surface into one of double
connectivity and gives two pieces of boundary: and let Q2 be another
cross-cut, not meeting the direction of Q1 anywhere but continuously
deformable into Qlt so that it also changes the surface into one of double
connectivity with two pieces of boundary. Then, in each of these doubly
connected surfaces, any number of functions can be uniquely determined
which satisfy the general conditions, each of which assumes assigned
boundary values, that is, along the boundary of 8 and the new boundary,
and possesses an arbitrarily assigned modulus of periodicity.

The combination of these functions, by an alternate process similar to

that for the preceding case, leads to a unique function which has an

assigned modulus of periodicity for the cross-cut Qlw The conditions

which determine it are: (i), the general conditions: (ii), the values along

F- 28

434 CONDITIONS OF EXISTENCE [227.

the boundary of the given surface, (iii) the value of the modulus of
periodicity for the cross-cut, which resolves the surface into one of double
connectivity, and the modulus of periodicity for the cross-cut, which
resolves the latter into a simply connected surface, that is, by assigned
moduli of periodicity for the two cross-cuts necessary to resolve the
original surface S into one that is simply connected.

Proceeding in this synthetic fashion, we ultimately obtain the result I
that a real function u exists for a surface of connectivity 2p + 1 with a single
boundary, uniquely determined by the following conditions : —

(i) its derivatives within the surface are everywhere uniform, finite

and continuous, and they satisfy the equation V2u = 0 ;
(ii) it assumes, along the boundary of the surface, assigned values
which are always finite but may be discontinuous at a limited
number of isolated points on the boundary ;

(iii) the function within the surface is everywhere finite and, except at
the positions of cross-cuts, is everywhere Uniform and continuous :
the discontinuities in value in passing from one edge to another
of the cross-cuts are arbitrarily assigned real quantities.
Now the surfaces under consideration are of odd connectivity : the func
tion thus determinate is everywhere finite, so that no points need to be
excluded from the range of variation of the independent variable ; the single
boundary of the closed surface can be made a point. The boundary value
is then a value assigned to the function at this point*; it may be depen
dent upon a value assigned to w at some point, in order to obtain the |
arbitrary additive imaginary constant in w subject to which it is dependent
upon u. Hence we infer that real functions exist on a Riemanns surface,
finite everywhere on the surface and uniquely determined by their moduli of
periodicity at the cross-cuts, which moduli are arbitrarily assigned real
quantities. It will be proved that the moduli cannot all be zero (§ 231).

228. The following important proposition may now be deduced : —
Of the real functions, which satisfy the general conditions and are finite
everywhere on the Riemanns surface, and are determined by arbitrarily
assigned modidi of periodicity, there are 2p and no more that are linearly
independent of one another ; and every other such function can be expressed,
except as to an additive constant, as a linear combination of multiples of these
functions with constant coefficients.

Taking into account only real functions, which satisfy the general
conditions and are everywhere finite, we can obtain an infinite number of
functions by assigning arbitrary moduli of periodicity.

* Or, if we please, the constant value along the circumference of a small circle round the
point ; in the absence of the conditions of uniformity and continuity, the proposition VI. of
§ 220 does not apply to this case.

228.] LINEARLY INDEPENDENT FUNCTIONS 435

When one function MJ has been obtained, with co1>1} w1)2,..., «1)21> as its
arbitrarily assigned moduli, another function u., can be obtained with

m-2,l> M-2,2, •••>' «<>2,2p

as its arbitrarily assigned moduli of periodicity, which are not the moduli of
kiUi, where k^ is a constant. A third function u3 can then be obtained, with
MS,I, «3,2> •••, &>3,2p as its arbitrarily assigned moduli of periodicity, which are
not the moduli of k^ + kzu2, where ^ and k2 are constants ; and so on, provided
that the number of functions obtained, say q, is less than 2p. When q < 2p,
another function can be obtained whose moduli of periodicity are different
from those of 2 krur. But when q = 2p, so that 2» definite functions,

r=l

linearly independent of one another, have been obtained, it is possible to
determine constants klt k«, ...,k.2p, so that

r=l

(for m = 1,2,..., 2p), where ni} O2, . . . , fL2p are arbitrary constants.

Let U be the potential function, which satisfies the general conditions
and is finite everywhere on the surface and is determined by the arbitrarily
assigned constants fllt O2, ..., £l2p; then the function

2p

r=l

has all its moduli of periodicity zero, it is everywhere finite and, because its
moduli are zero, it is uniform and continuous everywhere on the surface. It
is therefore, by § 220, a constant ; and therefore

2p

U = 2, krur + A ,

• . r=l

proving the proposition.

229. The only remaining condition of § 214 to be considered is the
possible possession, by the function u, of infinities of assigned forms, at
assigned positions on the surface.

Let the infinity at a point on the surface, where z is equal to cr, be
represented by the real part of <f> (z, cr), where

and let this real part be denoted* by $</>(>, cr) ; then u-3t$(z, cr) has no
infinity at z = cr. Proceeding in the same manner with the other assigned
infinities at all the assigned points, we have a function

, cr),

The form of 0 (z, cr) implies that the series giving the infinite terms has negative integral
exponents ; the case, in which the exponents are proper fractions so that the point is a branch
point, is covered by the transformation of § 223 when the modified form of 0 explicitly satisfies
the tacit implication as to form.

28—2

436 CLASSES OF FUNCTIONS [229.

which has no infinities on the surface. Its derivatives everywhere (save at
branch-points) are finite, uniform and continuous and satisfy the equation
V2« =0. If T be a typical representation of the assigned boundary values
of u and <E> be the corresponding typical representation of the assigned
boundary values of S 9t</> (z, cr), then T - <t> is a typical representation of

r=l

the boundary values of U.

The moduli of periodicity of U may arise through two sources : (i)
arbitrarily assigned real moduli of periodicity at the 2p cross-cuts of the
canonical system (§ 181), that are necessary to resolve the original surface into
one that is simply connected : (ii) the various moduli Dft (27riBr), arising from
the infinities cr in the surface, the occurrence of which infinities renders these
additional moduli necessary for the various additional cross-cuts that must be
made before the surface can be resolved. Then U has all these moduli as its
moduli of periodicity : it is finite everywhere on the surface and, except for its
moduli of periodicity, it is uniform and continuous on the surface ; hence it is
a function uniquely determinate, which is a constant if all the moduli be zero.

It therefore follows that the determination of u is unique, that is, that a
real function u on the Riemanris surface is determined by the general conditions
at all points on the surface except infinities, by the assignment of specified forms
of infinities at isolated points, and by the possession of arbitrarily assigned
moduli of periodicity at the cross-cuts which must be made to resolve the
surface into one that is simply connected. And, when all the moduli are zero,
the real function u is uniform.

Now w, = u + iv, is determined by u save as to an arbitrary additive
constant. Hence, summarising the preceding results, we infer the existence
of the following classes of functions on the surface : —

(A) Functions which are finite everywhere on the surface and, except

at the lines of the cross-cuts which suffice to resolve the surface
into one that is simply connected, uniform and continuous ;
and which have, at these cross-cuts, moduli of periodicity,
the real parts of which are arbitrarily assigned constants : —

(B) Functions which have a limited number of assigned singularities

(either algebraical, or logarithmic, or both) at assigned isolated
points, and which otherwise have the characteristics of the
functions defined in (A).

The existence of the various kinds of functions, considered in the preceding
chapter in connection with a special form of Riemann's surface, will now be
established for any given surface.

CHAPTER XVIII.

APPLICATIONS OF THE EXISTENCE-THEOREM.

230. WE proceed to make some applications of the existence-theorem as
established in the preceding chapter in connection with any Riemann's surface,
that is supposed given geometrically in an arbitrary way ; and we shall first
consider it in relation with the functions usually known as Abelian trans
cendents.

The existence of various classes of functions of position has been established.
Let functions which, satisfying the general conditions, are finite everywhere on
the Riemann's surface and have assigned moduli of periodicity at the 2p
cross-cuts, be called functions of the first kind, in analogy with the nomen
clature of §§ 205 — 211 ; let functions which, satisfying the general conditions,
have assigned algebraical infinities on the Riemann's surface and have
assigned moduli of periodicity at the 2p cross-cuts, be called functions of
the second kind; and let functions which, satisfying the general conditions,
have assigned logarithmic and algebraical infinities* and have assigned moduli
of periodicity at the 2p cross-cuts as well as the proper moduli in connection
with the logarithmic infinities, be called functions of the third kind. These
classes of functions evidently contain the integrals of the respective kinds
which arise through algebraical functions.

First, let P and Q be two functions of x and y, the derivatives of which
are finite, uniform and continuous at all points (except possibly branch-points)
on the given Riemann's surface and satisfy the equation V'2u = 0. Let the
functions themselves be finite and, except at cross-cuts, uniform and
continuous on the surface: and let their moduli of periodicity be Al,...,
Ap, B1} . .., Bp\ AI, ..., Ap, BI, ..., Bp', for the cross-cuts au ..., ap, b1} ...,bp
respectively, the moduli for the cross-cuts c being zero. (If P and Q should
have infinities on the surface, as will be the case in later applications, so that
in their vicinity portions of the surface are excluded, thereby requiring other
cross-cuts for the resolution of the surface into one that is simply connected,
other moduli will be required ; but, in the first instance, P and Q have
merely the 2p assigned moduli.)

When the surface is resolved by the 2p cross-cuts into one that is simply

* The logarithmic infinities must be at least two in number, by § 210.

438

PROPERTIES OF A SYSTEM

[230.

connected, the functions P and Q are uniform, finite and continuous over
that resolved surface. Proceeding as in § 16 and § 216, we have

where the double integrals extend over the whole area of the resolved

surface, and the single integrals extend positively round

the whole boundary. This boundary is composed of a

single curve, composed of both edges of each of the

cross-cuts ; and the positive directions of the description

are indicated in the figure, at a point of intersection of

two cross-cuts.

As explained in § 196, the negative edge of the cross
cut ar is GE and the positive edge is DF ; the negative
edge of the cross-cut br is EF and the positive edge is CD. Then we have

PD-PF=Pc-PE = Br, PF-PE = PD-Pc=Ar;
and similarly for the function Q.

Consider the integral jPdQ, taken along the two edges of the cross-cut
ar : let P_ and P+ denote the functions along the negative and the positive
edges respectively, so that P+ -P_ = Ar. The value of the integral for the
two edges is

Fig. 82.

I)

P+dQ, taken in the direction F. . .D

fM

+ P-dQ, taken in the direction G...E
J c

rD

= (P+ — P_) dQ, taken in the direction F...D

J F

= Ar !DdQ = Ar (QD - QF} = ArBr'.

J F

Similarly, when the value of the integral for the two edges of the cross-cut br

is taken, we have

re

P+dQ, taken in the direction D... C
J D

ff

P-dQ, taken in the direction E...F

J E

— PJ)dQ, taken in the direction D...C
Q = Br (Qc -QD) = - BrAr'.

+

J

rc

D

= B

230.] OF MODULI 439

And the value of the integral for the combination of the two edges of any
cross-cut c is zero.

Hence summing for the whole boundary of the resolved surface, we have

jPdQ= ^ (ArBrf - BrAr'\
and therefore

subject to the assigned conditions.

This theorem is of considerable importance : and the conditions, subject
to which it is valid, permit P and Q (or either of them) to be real or complex
potential functions of a; and y or to be a function of z.

231. As a first application, let P and Q be real potential functions such
that P + iQ is a function of z, say w, evidently a function of the first kind.
Let its moduli for the cross-cuts be

(Os+ivs at as, for s = l. 2, ..., p ;
and Wg' + ivg' at bs, for s=l, 2, ...,p.

Since P + iQ is a function of x + iy, we have, by §§ 7, 8,

dx dy ' dy dx '
The double integral then becomes

3y + 1 ^

which cannot be negative, because P is real ; it is a quantity that is positive
except only when P (and therefore w) is a constant everywhere. In the
present case

p
so that 2, (atfVf — (or'vr) is always positive. Hence :

r=l

If a function w, everywhere finite on a Riemanris surface, have o)s + ivs at
a, (/or s=l, 2, ...,p) and to,' +ivs' at bs (for s=l,2, ...,p) as its moduli,
the cross-cuts a and b being the 2p cross-cuts necessary to resolve the surface
into one that is simply connected, then

p

2, (a)rvr' —
r=l

is always positive, unless w is a constant : and then it is zero,
This proposition has the following corollaries.

COROLLARY I. A function of z of the first kind cannot have its moduli of
periodicity for alf ..., ap all zero.

440 PROPERTIES [231.

For if all these moduli were to vanish, then each of the quantities wr and
each of the quantities vr would be zero: the sum 2 (a)rvrf — wr'vr) would
then vanish, which cannot occur unless w be a constant.

COROLLARY II. A function of z of the first kind cannot have its moduli of
periodicity for b1} ..., bp all zero; it cannot have its moduli of periodicity all
purely real, or all purely imaginary, or some zero and all the rest either
purely real or purely imaginary.

The different cases can be proved as in the preceding Corollary.

Note. One important inference can at once be derived, relative to
functions of the first kind that have only two moduli of periodicity,
flj and I12.

Neither of the moduli may vanish; for if one, say ni} were to vanish,
then w/fi2 would be a function having one modulus zero and the other unity.

The ratio of the moduli may not be real. If it were real, then w/flj would
be a function having one modulus unity and the other real. Both of these
inferences are contrary to Corollary II. ; and therefore the ratio of the two
moduli is a complex constant, the real part of which may vanish but not the
imaginary part.

The association of this result with the doubly-periodic functions is
immediate.

Ex. Shew that, if two functions of the first kind have the same moduli of periodicity,
their difference is a constant : and that, if W be a value, at any point of the surface,
of a function of the first kind with moduli o^, w2, ..., co2p, all the functions of the first
kind, which have those moduli, are included in the form

W+ 2 mra>r+A,

r=\

where the coefficients m are integers and A is a constant.

232. As a second application, let P be a function of z and Q also a
function of z ; evidently, with the restriction of the proposition, P and Q
must be functions of the first kind, when no part of the surface is excluded
from the range of variation of z. Then

.dP dP .dQ dQ

fl . • Q _V ^_

dx dy ' dx dy '
so that at every point on the surface we have

dx dy dx dy
Consequently the double integral

fdP^dQ_dQdP

$x dy dx dy

232.] OF MODULI 441

and therefore, if a function of the first kind have moduli A1} ..., Ap, B1} ...,BP,
and if any other function of the first kind have moduli A^, ..., Ap, £/, ..., Bp'
at the cross-cuts a and b respectively, then

I (ArBr' - BrAr') = 0.

r=l

233. Next, let Q be a function of z of the first kind, as in the preceding
case ; but now let P be a function of z of the second kind, so that all its
infinities are algebraical. The points where the function is infinite must be
excluded from the surface : a corresponding number of cross-cuts will be
necessary for the resolution of the surface into one that is simply connected.
The modulus of periodicity of P for each of these cross-cuts is zero, (as in Ex. 8
of § 199, which is an instance of a function of this kind), no additional
modulus being necessary with an algebraical infinity.

Then over the resolved surface, thus modified, the functions P (z) and
Q (z) are everywhere uniform, finite and continuous : and therefore, as
before

ftP dQ dQ dP\

57 — s2 :^- 1 dxdy = fPdO,

dy dx dyJ

the double integral extending over the whole of the resolved surface and the
single integral extending round its whole boundary. But, at all points in
the resolved surface, we have

8P8Q_8Q3P = ()
dx dy doc dy

and therefore, as before, the double integral vanishes. Hence fPdQ, taken
round the whole boundary, vanishes.

The boundary is made up of the double edges of all the cross-cuts a, b,
and those, say I, which are introduced through the infinities, and of the small
curves round the infinities.

As in § 230, the value of the integral for the two edges of ar is A rBr' ;
and its value for the two edges of br is - BrAr'. The value of the integral
for the two edges of any cross-cut I is zero, because the subject of integration
is the same along the edges which are described in opposite directions.

To find the value round one of the small curves, say that which encloses
an infinity represented analytically by a value cs of z, we take, in the imme
diate vicinity of cg,

where p(z- cs) is a converging series of positive integral powers of z - cg. In
that vicinity, let

Q = Qs + (z - c») Qs + higher powers of z - cs,

442 RELATIONS BETWEEN MODULI [233.

so that Qs is dQ/dz for z — cs ; thus

dQ = (Qg + positive powers of z — cs) dz.
Hence along the small curve

+q(z-c.)dz,

Z C

where q(z — Cg) is a converging series of positive integral powers of z — cs.
The value of the integral round the curve is 27riHsQs'.

Summing these various parts of the integral and remembering that the
whole integral is zero, we have

I (ArBr' - BrAr'} + 27ri2HsQs' = 0,

r=l

there being as many terms in the last summation as there are simple
infinities of P.

The equation

v

2 (ArBr' - BrAr')

r=l

is the relation which subsists between the moduli A', B' of a function Q(z) of
the first kind and the moduli A, B of a function P (z) of the second kind,
all the infinities of which are simple.

The simplest illustration is furnished by the integrals that were considered in Ex. 6
and Ex. 8 of § 199.

Let P be the function of Ex. 8, usually denoted by E(z), being the elliptic integral
of the second kind ; it is infinite for z = oo in each sheet. In the upper sheet we have,
for large values of |z ,

P = E(z) = kz (l+positive integral powers of -);
\ z/

and for the same in the lower, we have

P = E(z)~ -kz ( l+positive integral powers of -J.

Let Q be the function of Ex. 6, usually denoted by F(z\ being the elliptic integral
of the first kind, finite everywhere. We easily find, for large values of z\ in the upper
sheet, that

dQ — dF(z}~jr^ ( l+positive integral powers of -)<&,
KZ \ z)

and, for large values of \z in the lower, that

dQ = dF (z} = — T-2 (l+positive integral powers of -\dz.

KZ \ Z J

Then for large values of \z in the upper sheet, we have

PdQ = — [ l + positive integral powers of -)
z \ ZJ

= r (l + positive integral powers of z1),

233.] NUMBER OF FUNCTIONS OF FIRST KIND 443

where zz' = 1 ; and we may consider the Riemann's surface spherical. Hence the value
round the excluding curve in the upper sheet is -2iri.

Similarly the value round the excluding curve in the lower sheet is - 2iri.

Now Al and B±, the moduli of P, are 4E and 2i(K'-E') respectively; AJ and B{, the
moduli of Q, are 4/f and 2iK' respectively. Hence

4E. 2iK' - 4K. 2i (K1 - E') - 4»rt = 0,
leading to the Legendrian equation

234. Before proceeding to the relations affecting the moduli of periodicity
of functions of the third kind, we shall make some inferences from the
preceding propositions.

It has been proved that functions of the first kind, special examples of
which arose as integrals of algebraic functions, exist on a Riemann's surface.
They are everywhere finite and, except for additive multiples of the moduli,
they are uniform and continuous ; and when, in addition to these properties,
the real parts of their moduli of periodicity are arbitrarily assigned, the
functions are uniquely determinate. Hence the number of such functions is
unlimited : they are, however, subject to the following proposition: —

The number of linearly independent functions of the first kind, that exist on
a given Riemann's surface, is equal to p ; where '2p + 1 is the connectivity of
the surface. And every function of the first kind on that surface is of the

P
form (7+2 cqWg, where G is a constant, the coefficients c1( ..., cp are constants,

2 = 1

and w1} ..., wp are p linearly independent functions.

Let q series of linearly independent real quantities, each series containing
2p non-vanishing constants, be arbitrarily assigned as the real parts of the
moduli of periodicity of functions of the first kind, which are thence uniquely
determined. Let the functions be w1} w2, ..., wq; and let the real parts of
their moduli be (0)1,1, a>i,2> •••> wi,ip), (&>2,i, &>2,2, •••» ^a.^p). •••> (<*>q,i> wq,i, • ••,&>g)2p).
The modulus of wr at the cross-cut Gm has its real part denoted by wr>m:
when the modulus is divided into real and imaginary parts, let it be

If any set of q arbitrary complex constants be denoted by d, ..., cq, where
cs is of the form «s + t'/3s, then, at the cross-cut Gm, the real part of the

q q^ • i

modulus of 2 crwr is the real part of S cr (&>r>m + i(0'r,in), that is, it is equal to

r=l r=l

holding for m= 1, 2, ..., 2p and therefore giving 2p expressions in all.

Now let a series of real arbitrary quantities A1} A.2, ..., A2p be assigned as
the real parts of the moduli of periodicity of a function of the first kind,

444 INDEPENDENT FUNCTIONS OF FIRST KIND [234.

which is uniquely determined by them ; and consider the equations

A! = «!<»!,! + a2&>2)1 + . . . + aqo)Qtl - &G/M - £Xa,i - ... - Pqto'q,i }

yp = !!, 2p

First, let q<p: the 2g constants a and /3 can be determined so as to make
the right-hand sides respectively equal to 2q arbitrarily assigned constants A.
The right-hand sides of the remaining equations are then determinate con
stants ; and therefore the remaining equations will not be satisfied when the
remaining constants A are arbitrarily assigned.

The function determined by the moduli A has some of its moduli different
from those of the function Sew, when q<p; hence, when q functions w1} ...
..., wq, where q <p, have been obtained, we can obtain another function, and
so on ; until q=p-

But, when q=p, then the foregoing *2p equations determine the quantities
a and /3, whatever be the quantities A. Let W be the function of the first
kind, determined by the constants A as the arbitrarily assigned real parts of

its moduli of periodicity : then

v
W — 2 csws,

s=l

where the coefficients c are constants, has the real parts of all its moduli
of periodicity zero : it is therefore, by Cor. II. § 231, a constant, so that

W = G^U! + . . . + CpWp + C,

where C is a constant. Therefore the number of linearly independent
functions of the first kind is p ; and every function of the first kind is of

the form

p
(7+2 csws.

s = l

It has been assumed in what precedes that the determinant of the quanti
ties <o and a)' does not vanish. This possibility is not excluded merely by the
arbitrary choice of the quantities &> ; because the quantities o>' are determined
for w, and w is dependent on v. If, however, the determinant should vanish,
then, by taking the quantities a and /3 proportional to the minors of w and w
respectively in the determinant, all the quantities

can be made to vanish. These quantities are the real parts of the moduli of

p
periodicity of Z csws which, because they all vanish, is a constant, that is, the

s = l

quantities ws are not linearly independent of one another — an inference
contrary to their construction. Hence the determinant of the quantities to does
not vanish.

234] NORMAL FUNCTIONS OF FIRST KIND 445

Note. It may be remarked, in passing, that each function w, being of the
first kind, gives rise to two real potential functions, which are everywhere
finite and have moduli of periodicity at the cross-cuts : one of the functions
being the real part of w, the other arising from its imaginary part.
Hence from the p linearly independent functions of the first kind, there are
altogether 2p linearly independent real potential functions. This number is
the same as the total number of real potential functions considered in § 228 :
hence each of them can be expressed as a linear function of the members of
that former system, save possibly as to an additive constant. Conversely, it
follows that linear combinations of the members of that former system can be
taken in pairs, so as to furnish p (and not more than p) linearly independent
functions of z of the first kind.

235. The functions so far obtained are very general : it is convenient to
have a set of functions of the first kind in normal forms. The foregoing
analysis indicates that linear combinations of constant multiples of the
functions, being themselves functions of the first kind, are conveniently
considered from the point of view of their moduli of periodicity: and the
simpler the aggregate of these moduli is, the simpler will be the functions
determined by them. -Some conditions have been shewn (§ 231) to attach
to the aggregate of the moduli for any one function of the first kind, and a
condition (§ 232) for the moduli of different functions; these are the con
ditions that limit the choice of linear combinations.

Let clwl + . . . + CpWp be a linear combination of the functions w1 , . . . , wp
which have <wrl,..., wrp (r=l,...,p) as the moduli of periodicity for the
cross-cuts «!,..., ap. Then A, where A is the determinant

A = &>n, <«12, , a)lp

apl> Wp2y ; Wpp

cannot vanish : for otherwise by taking constants c1} ..., cp proportional to the

first minors, we should obtain a function £ cg'Wg, having all its moduli for the

*=i

cross-cuts a1, ..., ap zero and therefore, by § 231, merely a constant, so that
wl , . . . , wp would not be linearly independent. Hence A does not vanish.

Next, we can choose constants c so that the moduli of periodicity
vanish for the function 2 cswg at all the cross-cuts a, except at one, say ar,

s = l

and that there it has any assigned value, say iri. For, solving the
equations,

0 = d«M + c2&>S)2 + ... + CPO)S!P, (for s > r=I, 2, ...,p);
TTI — d&V, ! + C.xw,.)2 + . . . + cpo)rtp,

446

NORMAL FORM OF FUNCTIONS

[235.

the determinant of the right-hand side does not vanish, and the constants c,
say cr>1, cr>2, ..., crjp, are determinate. The function cr>iW>i + cTi2w2 + . . . + cr>pwp,
say Wr, then has its moduli zero for al,..., ar_1} ar+1,..., ap : it has the
modulus iri for ar; it has moduli, say Br>1> ..., Br>p at 61} ..., bp respectively.
And the function is determinate save as to an additive constant.

This combination can be effected for each of the values l,...,p of r :
and thus p new functions will be obtained. These p functions are linearly
independent : for, if there were a relation of the form

GlWl-\-G^W»+ +GpWp = constant,

8

the modulus of the function z CrWr at the cross-cut as should be zero

r=l

because the function is a constant ; and it is Gairi, so that all the coefficients
G would be zero.

The functions W, thus obtained, have the moduli : —

*>!

W

w

•t

*

These functions are called normal functions of the first kind : they are a
complete system linearly independent of one another, and are such that every
function of the first kind is, except as to an additive constant, a linear com
bination of constant multiples of them.

The quantities B are not completely independent of one another. Since
Wj, Wj> are functions of the first kind we have, by § 232,

p

r=l

which, for the normal functions, takes the form

TriBjj' — TriBj'j — 0,

that is, BJJ> = BJ'J. Hence the moduli B with the same integers for suffix are
equal to one another.

This is a first relation among the moduli. Another is given by the
following theorem : —

235.] OF THE FIRST KIND 447

Let Bmjn = pm.jn +i<rm.!n, (so that pm,n = pn,m, and <rmin = o-n>m): then, if
clt ..., Cp be any real quantities, the expression

pnCi2 + 2/012C1C2 + /DooC22 + . . . + pppCp*,

is negative, unless the quantities c vanish together.

The function c1W1+c2W2+ ... + CPWP + G is a function of the first kind
with moduli (say) wr + ivr at ar, where r = 1, ..., p, and moduli to/ + ivs' at bs,

p
where s = l, ..., p. Then, by § 231, the sum 2, (wrvr' — wr'vr) is positive,

r=\

except when the function is a constant, that is, except when c:, ..., cp all

vanish. But

o)r + ivr — cr7ri,
so that ayr = 0, vr = 7TCr ; and

ft>/ 4- ivg = cj$l>s + c252jS + ... + CpBptS,
so that &>/ = Cj/Oj^ + C2p2,s + ... + cppptS.

p
Hence the sum i — cr7r fafar + C2p2i?. + . . . + cppptr)

r=l

p P
is positive and therefore the sum 2 Z /?,-«£,. cs is negative. This (with the

?•=! s=l

property pmw = pnm) is the required result.

These properties of the periods, all due to Riemann, are useful in the
construction of the Theta-Functions.

For the ordinary Jacobian elliptic functions in which p = 1, there is only one
integral which is everywhere finite : its periods are 4>K, 2iK'. To express it
in the normal form, we take cF (z), choosing c so that the period at a1 is

7T?/

purely imaginary and =TTI; hence c= \TT, and the normal integral is

T^X\_

iriF(z)
4K '

TrK'

The other period of this function is — --^ , which, when k is real and less than

CxL

unity, is a negative quantity ; it is the value of pn and satisfies the condition
that pnCj2 is negative for all real quantities c.

236. It has been proved that functions exist on a Riemann's surface,
having assigned algebraical infinities and assigned real parts of its moduli
of periodicity, but otherwise uniform, finite and continuous. The simplest
instance of these functions of the second kind occurs when the infinity is an
accidental singularity of the first order.

Let the single infinity on the surface be represented by z = c: let Ec(z)
be the function having £=cas its algebraical infinity, and having the real
parts of its moduli of periodicity assigned. If Ec' (z} be any other function
with that single infinity and the real parts of its moduli the same, then

448 NORMAL FUNCTION OF SECOND KIND [230.

Ec(z) — Ec(z) is a function all the real parts of whose moduli are zero; it
does not have c for an infinity and therefore it is everywhere finite : by § 231, it
is a constant. Hence an elementary function of the second kind is determined,
save as to an additive constant, by its infinity and the real parts of its moduli.

Again, it can be proved, as for the special case in § 208, that an elementary
function of the second kind is determined, save as to an additive function of
the first kind, by its infinity alone : hence, if E (z) be any elementary function,
having its infinity represented by z = c, we have

where \, ..., \p, A are constants, the values of which depend on the special
function chosen. Let Ec(z) have iriG^ ..., TriCp for its moduli at the cross
cuts «!,..., ap respectively : and let the function E(z) be chosen so as to have
all its moduli at al , . . . . ap equal to zero : then Ar = - Cr and E (z) is given by

Ec(z)-C,Wl-...-CpWp + A.

The special function of the second kind, which has all its moduli at the cross
cuts a-i, . . . , ap equal to zero, is called the normal function of the second kind.
It is customary to take unity as the coefficient of the infinite term, that is,
the residue of the normal function.

This normal function is determined, save as to an additive constant, by its
infinity alone. For if E (z) and E' (z) be two such normal functions, the function

E(z)-E'(z}

is finite everywhere; its moduli are zero at a1} ..., ap ; hence (§ 231) it is a
constant.

Normal functions of the second kind will be used later (§ 241) in the
construction of functions with any number of simple infinities on the surface.

Let the moduli of this normal function E (z) of the second kind be Bl, . . . ,
Bp for the cross-cuts 61} ..., bp. Then applying the proposition of § 233 and
considering the integral fEd Wr , we have A1 = . . . = Ap = 0 ; also

AI — ... = A r_l = A r+1 = . . . = Ap = 0,
and Ar' = iri. The relation therefore is

. fdWr\

- Br7ri + 2-m —r-? = 0,
V dz /,,j

where, in the immediate vicinity of z = c,

E(z} = —- + p(z-c\

Z ~~ G

p being a converging series of positive powers. Thus

dW
or, as -~ is an algebraical function (§ 241) on the surface, the periods of a

CIZ

236.] NORMAL FUNCTION OF THIRD KIND 449

normal function of the second kind at the cross-cuts b are algebraical functions
of its single infinity.

In the case of the Jacobian elliptic integrals, the integral of the second kind has at
z= oo an infinity of the first order in each sheet (Ex. 8, § 199). The moduli of this integral,
denoted by E(z\ are 4E and %i(K' — E') for a^ and b± respectively; hence the normal
integral of the second kind is

E(z)-ERF(z),

F(z) being the (one) integral of the first kind. This is the function Z(z)\ its modulus is
zero for «x, and for bl it is

which is ~(KK'-E'K-EK'\ that is, it is -^.
A A

237. The other simple class of function which exists on a Riemann's
surface with assigned infinities and assigned real parts of its moduli is that
which is represented by the elementary integral of the third kind. It has
two points of logarithmic infinity on the surface*, say Pj and P2; let these
be represented by the values d and c2 of z. On division by a proper constant,
the function, which may be denoted by II12, takes the forms

- log (z - Cj) + pl (z - d), + log (z - c2) + p.2 (z - c2),

in the immediate vicinities of Pj and of P2 respectively, where pl and p.2 are
converging series of positive integral powers.

The points P1 and P2 can be taken as boundaries of the surface, as in
Ex. 7 in § 199. A cross-cut from P2 to P1 is then necessary for the resolution
of the surface : and the period for the cross-cut is 2?n', being the increase of the
function in passing from the negative to the positive edge of the cross-cut.

Then with this assignment of infinities and with the real parts of the
moduli at the cross-cuts alt ..., ap, 61} ..., bp arbitrarily assigned, functions TI12
exist on the Riemann's surface.

As in the case of the function of the second kind, it is easy to prove that
1 a function IT12 of the third kind is determined, save as to an additive constant,
by its two infinities and the assignment of its moduli : and that it is deter
mined, save as to an additive function of the first kind, by its infinities alone.
Among the infinitude of elementary functions of the third kind, having
the same logarithmic infinities, a normal form can be chosen in the same
manner as for the functions of the second kind. Let II12 be an elementary
function of the third kind, having P1 and P2 for its logarithmic infinities : let
its moduli of periodicity be 2iri for the cross-cut P^P,; TtiCl, ..., iriCp for
fli, ..., ap respectively; and other quantities for blt ..., bp respectively. Then

* The representation of a single point on the Riemann's surface by means solely of the value of
z at the point will henceforward be adopted, without further explanation, in instances when it can
not give rise to ambiguity. Otherwise, the representation in full detail of statement will be adopted.

F. 29

450 MODULI OF NORMAL ELEMENTARY [237.

is an elementary function of the third kind, having zero as its modulus of
periodicity at each of the cross-cuts a1( ..., ap. This function is the normal
form of the elementary function of the third kind.

If OTJ/ and -5712 be two normal elementary functions of the third kind with
the same logarithmic infinities and the same period 2-Tn at the cross-cut
PaPa> then

•37,0 —

is a function without infinities on the surface ; its modulus for PXP2 is zero,
and its modulus for each of the cross-cuts a1} ..., ap is zero ; and therefore it
is a constant. Hence a normal elementary function of the third kind is, save
as to an additive constant, determined by its infinities alone.

Ex. The sum of three normal elementary functions of the third kind, having as
their logarithmic infinities the respective pairs that can be selected from three points,
is a constant.

238. A relation among the moduli of an elementary function of the third
kind can be constructed in the same way as, in § 233, for the function of the
second kind.

Let the surface be resolved by the 2p cross-cuts c^, ..., ap, blt ..., bp and by
the cross-cut P^, joining the excluded infinities of an elementary function
II12 of the third kind. Let w be any function of the first kind ; then over the

resolved surface, we have

3TT12 dw 3II12 dw

dx dy dy dx

everywhere zero; and therefore JII12dw round the whole boundary of the
resolved surface is zero, as in § 233.

Let the moduli of II]2 be A,,..., Ap, B,,..., Bp, and those of w be
A/,..., Ap, BI, ... , Bp for the 2p cross-cuts a and b respectively.

The whole boundary is made up of the two edges of the cross-cuts a, the
two edges of the cross-cuts b, the two edges of the cross-cut P^ and the
small curves round Px and P2.

The sum of the parts contributed to JTI12 dw by the edges of all the cross
cuts a and b is, as in preceding instances,

I (ASBS'-AS'BS).

The direction of integration along P^ that is positive relative to the area
in the resolved surface is indicated by the arrows ; the p p

portion of JTI12 dw along the two edges of the cut is Q — ^ -Q

Fig. 83.

rc2 /v?a

= (ni2+ - ni2~) dw = 27ri dw =

J ct *6l

238.]

FUNCTION OF THE THIRD KIND

451

Lastly, the portion of the integral for the infinitesimal curve round P1 is zero,
by I. of § 24, because the limit of (z — Cj) II12 -y- for z = d vanishes, Pj being

assumed not to be a branch-point ; and similarly for the portion of the
integral contributed by the infinitesimal curve round P2.

As the integral JTI12 dw vanishes, we therefore have

I (ASBS' - AS'BS} + 2m [w (c2) - w (c,)} = 0,

which is the relation required.

The most important instance is that in which II12 is the normal elementary
function of the third kind (and then A1} A2, ..., Ap all vanish), and w is a
normal function of the first kind, say Wr (and then

A/ = m, Ai=Az'= ... = A'r-j. = A'r+l = . . . = Ap = 0).

Hence, if Br be the modulus at br of the normal elementary integral t/r12, we
have

so that the moduli of the normal elementary function of the third kind can be
expressed in terms of normal functions, of the first kind, of its logarithmic

discontinuities.

The important property of functions of the third kind, known as the
interchange of argument and parameter, can be deduced by a similar process.

Let IT12 be an elementary function with logarithmic discontinuities at
d and c2, with 2m as its modulus for the cross-cut CiC2, and with

as its moduli for the cross-cuts a1; ..., ap, bl} ..., bp; and let IT34 be another
elementary function with logarithmic discontinuities at cs and c4, with 2m as

its modulus for the cross-cut c3c4, and with
moduli for the cross-cuts alt ..., ap> blt ..., bp.
Then when the infinities are excluded and the
surface is resolved so that both ni2 and TT^
are uniform finite and continuous throughout
the whole surface, we have

**M"T <"\ TTT ^\"i— T ^NTT

^2=0,

/, •••, Bp as its

Fig. 84.

due dy "dx dy

everywhere in the resolved surface ; and therefore, as in the preceding
instances, fU^dYl^ round the whole boundary vanishes.

The whole boundary is made up of the double edges of the cross-cuts a
and the cross-cuts b, and of the configuration of cross-cuts and small curves
round the points. The modulus of both ni2 and of II^ for the cut AG is

29—2

452 INTERCHANGE OF ARGUMENT [238.

zero ; the modulus of IT12 for the cut C3c4 is zero, and that of TI^ for the cut
dC2 is zero.

The part contributed to JUlzdUM by the aggregate of the edges of the

cross-cuts a and 6 is 2 (A ,8, - AS'BS), as in preceding cases.
*=i

The part contributed by the small curve round Cj is zero, because the
limit, for z = d, of (z - Cj) II12 -, -- is zero ; similarly the part contributed by
the small curve round c» is zero.

The part contributed by the two edges of the cross-cut dca is

The part contributed by the two edges of the cross-cut AO is

o

the subject of integration does not change in crossing from one edge to the
other, and therefore this part is zero.

For points on the small curve round c3, we have

dU3i = -I- p (z - c3) dz,

z — c3

where p is a converging series of integral powers of z — cs : and therefore for
points on that curve

where q (z — c3) is a converging series of positive integral powers of z — c3.
Hence the part contributed to JTI12 dUu by the small curve round c3 in the
direction of the arrow, which is the negative direction for integration relative
to cs, is 27T* II12 (c3).

Again, for points on the small curve round c4, we have

dnM = — Z- +pi(z- c4) dz ;

Z — C4

proceeding as for c3, we find the part contributed to JU^dU^ by the small
curve round c4, which is negatively described, to be — 2?™' II 12 (c4).

Lastly, the sum of the parts contributed by the two edges of the cross-cut
C3c4 is

C*».

fet

J C3

^ --- j

dz dz

238.] AND PARAMETER 453

But though n34 has a modulus for the cross-cut c3c4, its derivative has not a
modulus for that cross-cut: we have dU.^/dz = dll^~/d2, and therefore the
last part contributed to /ni2 dH3i vanishes.

The integral along the whole boundary vanishes ; and therefore

I (ASB; - A;BS) + 2™ (n* (Ca) - nsl (Cl)} + 2^^, (c3) - 2™ii12 (c4) = o,
*=i

a relation between the moduli of two elementary functions of the third kind.
The most important case is that in which both of the functions are normal
elementary functions. We have A1} ..., Ap zero for cr12, and AI , ..., Ap' zero
for -5734 ; and the relation then is

«84 (CS) - OT.J4 (d) = «T12 (C4) - OT12 (C3),

which is often expressed in the form

P2 1 f4 j

dv?34= I aarjti

J c, •/ c3

the paths of integration in the unresolved surface being the directions of
cross-cuts necessary to complete the resolution for the respective cases.
Hence the normal elementary integral of the third kind is unaltered in value
l)ij the interchange of its limits and its logarithmic infinities.

239. From the simple examples, discussed in § 199 and elsewhere, it has
appeared that when a function w is defined as the integral of some function
of z, the integral being uniform except in regard to moduli of periodicity, a
process of inversion is sometimes possible whereby z becomes a function of w,
either uniform or multiform. But in all the cases, in which z thus proves to be
a uniform function, the number of periods possessed by w is not greater than
two ; and it follows, from §110, that, when w possesses more than two periods,
z can no longer be regarded as a function of w. In fact, w then loses its
property of being uniform by dependence upon a single variable.

A question therefore arises as to the form, if any, of functional inversion,
when w has more than two independent periods and when there are more
functions w than one.

Taking the most general case of a Riemann's surface of class p, let
w1} w.2, ..., wp denote the p functions of the first kind. Let there be q inde
pendent variables zlt...} zq, where q is not, of initial necessity, equal to p\
and, by means of any q of the functions of the first kind, say w1} .,,, iuq, form
q new functions, evidently also of the first kind and defined by the equations

vr = wr Oj) + wr (>2) +...+wr (zg),

where r = 1, 2. ..., q. We make the evident limitation that q is not greater
than p, which is justifiable from the point of view of functional inversion.
Then the functions vr are multiform on the surface with constant moduli of
periodicity; they have the same periods as wr, say coftl, wr^, ..., wr,y>-

The various values of wr (zm) differ by multiples of the periods : so that, if

PEOBLEM OF INVERSION [239.

wr(z>m) be the value for an exactly specified .s^-path (as in § 110), the value
for any other ,^-path is

This being true for each of the integers m= 1, 2, ..., q, it follows that, if

<?
ms= 2 nm>s, (s = l, 2, ..., 2p),

m=l
q

and if vr be the value of S wr (#w) for the exactly specified paths for z^ , . . . , zq,
m=l

then the general value of vr for any other set of paths for the variables is
vr + tn^a)rtl + mzwr»_ 4- . . . + mzpwrflp,

holding for r = l, 2, ..., q. The integers nmiS, and therefore the integers mg,
are evidently the same for all the functions v.

The reason which, in the earlier case (§ 110), prevented the function w from
being determinate as a function of z alone was, that integers could be deter
mined so as to make the additive part of w, dependent upon the periods, an
infinitesimal quantity. It is necessary to secure that this possibility be
excluded.

Let &>A!M = aA]/x + iySAj/lt, where the quantities a and /3 are real : then we
have to prevent the possibility of the additive portions for all the functions v
being infinitesimal. In order to reduce the additive part to an infinitesimal
value for each of the functions v, it would be necessary to determine integers
Wi, in.,, ..., m2p so that the 2q quantities

for r — 1, . . . , q all become infinitesimal.

If q be less than p, the 2p integers can be so determined. In that case,
the general possibility of functional inversion between the q functions v and
the q variables z would require that the quantities z are so dependent upon
the quantities v that infinitesimal changes in the latter, carried out in an
infinite variety of ways and capable of indefinite repetition, would leave the
quantities z unchanged. The position, save that we have q variables instead
of only one, is similar to that in § 110 : we do not regard the functions v as
having determinate values for assigned values of z1, ..., zq, but the values of
«!, ..., vq are determinate, only when the paths by which the independent
variables acquire their values are specified. And, as before, the inversion is
not possible.

If q be not less than p, so that it must in the present circumstances be
equal to p, then the 2p integers cannot be determined so that the 2p quanti
ties all become infinitesimal. They can be determined so as to make any
2p — 1 of the quantities become infinitesimal ; but the remaining quantity is

239.] INVERSION 455

finite as, indeed, should be expected, because the determinant of the constants
a and /3 is different from zero*.

If then there be^> variables zl, ..., zp, andp functions vl} ..., vp defined by

the equations

vr = wr (z^ + wr 0.) + . . . + wr (zp},

for r = 1, 2, . . . , p, then the values of the functions v for assigned values of the
variables z, whatever be the paths by which the variables attain these values,
are of the form

Vr + 7/ij Q)r>1 + W2&)ri2 + . . . + WiapOV.ap

for r = 1, 2, ..., p ; and it has been proved that the 2p integers in cannot be
determined so that all the additive parts, dependent upon the periods, become
infinitesimal. Hence the functions v1,..., vp are, except as to additive
multiples of the periods (the numerical coefficients in these multiples being
the same for all the functions), uniform functions of the variables zlt ..., zp;
and they are finite for all values of the variables. Conversely, we may regard
the quantities z as functions of the quantities v^, ..., vp, which have 2p sets of
simultaneous periods &>M, &)2)S,..., wpiS for s=l, 2,..., 2p : that is, the
variables z are 2p-ply periodic functions of p variables v1} ..., vp. This result
is commonly called the inversion-problem for the Abelian transcendents.
In effecting the inversion of the equations

dvj, = MI (z^ dzl + Wi (z2) dz2+ ... + w± (zp) dzp\

dvp = Wp (z^) dzl +wp(z2) dz2+ ... +wp (zp) dzp)

the actual form, which is adopted, expresses all symmetric functions of the
quantities z1} ..., zp as uniform functions of the variables, so that, if zl} z%, ... ,
zp be the roots of the equation

then-}- P!,..., Pp are uniform multiply-periodic functions of the variables
v1}..., vp. Consequently, all rational symmetric functions of z1} ..., zp are
uniform multiply periodic functions of v1} ..., vp.

Frequent reference has been made to the functions determined by the equation
w2- R(z)=w*- (z—a0) (z-al)...(z-a2p) = 0.

It has been proved that an integral of the form I — — dz is an integral of the first

kind, provided U(z) be an integral algebraical function of degree not higher than p — 1, and
that the otherwise arbitrary character of U(z) makes it possible to secure the necessary
p integrals by allowing the suitable choice of the coefficients. Weierstrass takes the
equations, which lead to the inversion, in the following form J : —

* The 2p potential-functions, arising from the p functions w, would otherwise not be linearly
independent.

t For further considerations see Clebsch und Gordan, Theorie der Abel'schen Functionen,
Section vi.

J Equivalent to that given in Crelle, t. lii, (1856), pp. 285 et seq.; it is slightly different from
the form adopted by him in Crelle, t. xlvii, (1854), p. 289.

456 ABELIAN FUNCTIONS [239.

The constants a are different from one another and can have any values : and it is
convenient to take

P(x) = (x-al}(x-a3)...(x-a2p.1),

Q (a?) = (x - Oo) (x - «2). ..(x-a2p_ 2) (x - a.2p),
so that P (x) Q(x) = R(x). If the coefficients a be real, it is assumed that

The equations which give the new variables are
dul =

and when integration takes place, the arbitrary constants are defined by the equations

ui> M2>---> uP=Q (with periods for moduli),

when zv 22,... , 2p=a1? «3)... , «2p-i respectively.

The p variables s are the roots of an algebraical equation of degree p, the coefficients in
which are (multiply-periodic) uniform functions of the variables u. The functions, arising
out of the equations in this form, are discussed* in Weierstrass's two memoirs, just
quoted.

Note 1. The results thus far established in this chapter are the basis of the theory of
Abelian functions. The development of that theory is beyond the range of the present
treatise.

So far as concerns the general theory, recourse must be had to the fundamental
memoirs of Abel, Jacobi, Hermite, Riemann and Klein, and to treatises, in addition to
those by Neumann and by Clebsch and Gordan already cited, by Prym, Krazer, Konigs-
berger and Briot.

Moreover, as our propositions have for the most part dealt with functions of only
a single variable, it is important in connection with the Abelian functions to take account
of Weierstrass's memoir f on functions of several variables.

Note 2. We have discussed only very limited forms of integrals on the Riemann's
surface : and any professedly complete discussion would include the theorem that $w'dz,
where w' is a general function of position on the surface, can be expressed as the sum of
some or all of the following parts : —

(i) algebraical and logarithmic functions;
(ii) Abelian transcendents of the three kinds;
(iii) derivatives of these transcendents with regard to parameters;
but such a discussion is omitted as appertaining to the investigations relative to Abelian
transcendents.

For the particular case in which the integral JVcfe is an algebraical function of 2, see
Briot et Bouquet, The'orie des fonctions elliptiques, (2me e"d.), pp. 218 — 221 ; Stickelberger,
Crelle, t. Ixxxii, (1877), pp. 45, 46; and Humbert, Acta Math., t. x, (1887), pp. 281—298,
by whom further references are given.

* Some of the results are obtained, somewhat differently, in a memoir by the author, Phil.
Trans., (1883), pp. 323—368.

t First published in 1886 ; Abhandlungcn aux der Functionenlehre , pp. 105 — 164.

240.] UNIFORM FUNCTIONS ON RIEMANN's SURFACE 457

240. There are functions belonging to class (B) in § 229, other than
those already considered. In particular, there are functions with assigned
infinities on the surface and with the real parts of all their moduli of
periodicity for the canonical system of cross-cuts equal to zero. But it
does not therefore follow that all the moduli of periodicity vanish ; in order
that their imaginary parts may vanish, so as to make the moduli of
periodicity zero, certain conditions would require to be satisfied.

We shall limit the ensuing discussion to some sets of these functions
with zero moduli, and shall assign the conditions necessary to secure that
the moduli shall be zero. We shall assume that all their infinities are
algebraical ; the functions are then uniform everywhere on the surface,
and, except at a limited number of isolated points where they have only
algebraical infinities, are finite and continuous. They are, in fact, algebraical
functions of z.

Two classes of these functions are evidently simpler than any others.
The first class consists of those which have a limited number, say m, of
isolated accidental singularities each of the first order and which are not
infinite at any of the branch-points ; the other class consists of those which
have no infinities except at the branch-points. These two classes will be
briefly discussed in order.

Let w be a uniform function having accidental singularities, each of the
first order, at the points c1, ..., cm and no other infinities ; and let the normal
function of the second kind, having cr for its sole infinity, be Zr. Then

where /31( ..., ftm are constants at our disposal, is a function, having infinities
of the same class and at the same points as w has ; the function is otherwise
finite everywhere on the surface and therefore, by properly choosing the
constants /3, we have the function

finite everywhere on the surface, so that it is a function of the first kind.

Now because its modulus vanishes at each of the cross-cuts a in the
resolved surface, it is a constant, so that

w = /3lZ1+ ...+/3mZm + /30.

dW .
The modulus of w is to vanish at each of the cross-cuts br. Let <f>r(z) = ~r-^ ,

so that </>,. (z) is an algebraical function on the surface : then assigning the
condition that the modulus of w at the cross-cut br shall vanish, we have

&</>,. (Cl) + Mr (C2) + . . . + /3m<f)r (Cm) = 0,

an equation which must hold for all the values r = 1,...,p.

When the quantities c represent quite arbitrary points, there must be
at least p -I- 1 of them ; otherwise, as the equations are independent of one
another, they can be satisfied only by zero values of the constants (B, a result

458 UNIFORM FUNCTIONS IN TERMS OF [240.

which renders the uniform function evanescent. If m > p, the equations
determine^ of the coefficients /3 linearly in terms of the remaining m—p:
when these values are substituted, the resulting expression for w contains
m—p + 1 constants, viz., the remaining m —p constants /3, and the constant
/30. The coefficient of each of the in —p constants /3 is a function of z, which
has p 4- 1 accidental singularities of the first order, p of which are common
to all the functions, so that w then is an arbitrary linear combination of
constant multiples of in — p functions, each of which possesses p 4- 1
accidental singularities and can be expressed in the form

Xl\t Zln, , •"pj *~P+f

01 (^l), 01 (Co), , 01 (C^>), 01 (Cp+r)

03 (C,)> 03 (C2), > 02 (Cp), 0a(Cp+r)

0j> \p\)t 0w\C:>)j > Vp\Pp)> TP \^)+r/

When the quantities c are not completely arbitrary, but arc such that
relations among them can be satisfied so as no longer to permit the preceding
forms to be definite, we proceed as follows.

The most general way in which the preceding forms cease to be definite
is by the dependence of some of the equations

&0r (Ci) + /320r (C,) + • • • + &»0r (Cm) = 0

on the remainder. Let q of them, say those given by r = 1, ..., q, be de
pendent on the remaining p - q, so that q > 0 < p : then the conditions of
dependence can be expressed by equations of the form

<f)r (cn) = Alirtf>q+1 (Cn) + Ai>r 03+2 (Cn) + . . . + Ap_q>r(f>p (Cn)

for r = 1, 2, . . . , q and n = 1, 2, . . . , m.

The functions of the first kind W, through which the functions 0 are
derived, are a complete set of normal functions : when any number of them
is replaced by the same number of independent linear combinations of some
or all, the first derivatives are still algebraical functions. We therefore
replace the functions Wl} W2,..., Wq by wl} w,,..., wq, where

Wr = Wr — AI!T W 3+! — ^J-2,r "9+2 • •• -"-p—q,r " p

for r = 1, 2, ...,q, so that, for all values of z,

Hence the functions ^>l, <£2, ... , <&q vanish at each of the points d, c,, ..., cm.
The original system of p equations in 01; ..., 03, 03+lf..., 0^, when
made a system of equations in <E>X) ..., <&q, 03+i, ••-, 0^ is equivalent to

o-i/vnl

for r = 1, . . . , ? and s = q + 1, . . . , p. The first q of these are evanescent ; and
therefore their form is the same as if we had initially assumed that each of

240.]

NORMAL FUNCTIONS OF THE SECOND KIND

459

the functions 01( ... , (j)q vanished for each of the points z = cl} ..., cm, the two
assumptions being in essence equivalent to one another on account of the
property of linear combination characteristic of functions of the first kind.

Suppose, then, that q of the functions 0, derived through functions
of the first kind, vanish at each of the points c1} ..., cm; the number of
surviving equations of the form

&0r (CO + &0r (C3) + . . . + /3™0, (Cm) = 0

is p — q, and they involve m arbitrary constants ft. Hence they determine
p — q of these constants, linearly and homogeneously, in terms of the other
m-p + q. When account is taken of the additive constant ft0, then* the
function w contains m—p+q+l arbitrary constants; and it is a linear
combination of arbitrary multiples of m—p + q functions, each having p — q + l
accidental singularities of the first order, p — q of which are common to all
the functions in the combination.

The functions under consideration, being linear combinations of normal
functions Z of the second kind, have no infinities except at the accidental
singularities ; the branch-points of the surface are not infinities. And it
appears, from the theorem just proved, that there are functions having
only p — q + 1 accidental singularities, each of the first order, so that the total
number is less than p+l. A question therefore arises as to what is the
inferior limit to the number of accidental singularities that can be possessed
by a function which is uniform on the Riemann's surface and, except at these
accidental singularities, is everywhere finite and continuous on the surface.

Let it be denoted by /JL ; then the p equations

&<MC0 + • • • + /^(O = 0,

for r = 1, 2, ..., p, must determine /A — 1 of the constants ft in terms of the
remaining constant ft, say, B ; and the function thence inferred contains two
constants, viz., the surviving constant ft and the additive constant, its form
being

A+B

zti

01 (Ci),

Among the points c1; c.,, ..., c^, the relations

* This is usually known as Eiemann-Roch's Theorem. It is due partly to Riemann and
partly to Roch ; see references in § 242.

460 RIEMANN-ROCH'S THEOREM [240.

for r=0, l,...,p-p, must be satisfied, that is,p-fj,+ l relations must be
satisfied*.

Since there are /JL points c among which p — /j, + 1 relations are satisfied it
follows that the number of surviving arbitrary constants c is, in general, equal
to p — (p — /m + 1), that is, to 2/A — p — 1. These occur as arbitrary constants
in the inferred function, independently of the two constants A and B : so that
the number of arbitrary constants, in the function with p accidental singu
larities, is 2/4 — p — 1 + 2, that is, 2/A - p + 1.

Again, the number of infinities of a uniform function of position on a
Riemann's surface is equal to the number of its zeros (§ 194), and also to the
number of points where it assumes an assigned value ; and all these pro
perties are possessed by any function, with which w is connected by any
lineo-linear relation. If u be one such function, then another is

au + b

w = - — j ,
u — d

where a, b, d are arbitrary constants ; and therefore w contains at least
three arbitrary constants, when it is taken in the most general form that
possesses the assigned properties.

But it has been shewn that the number of independent arbitrary con
stants in the general form of w is 2/i — p + l. This number has just been
proved to be at least three, and therefore

2yU, - p + 1 > 3,

or fjt ^ 1 + \p.

Thus the integer equal to, or next greater than, I + ^p is the smallest number
of isolated accidental singularities that an algebraical function can have on a
Riemann's surface, on the supposition that it has no infinities at the branch
points^.

241. The other simple class of uniform functions on a Riemann's
surface consists of those which have no infinities except at the branch
points of the surface.

They will not be considered in any detail : we shall only briefly advert
to those which consist of the first derivatives of functions of the first kind.
This set is characterised by the theorem : —

These functions (ft (z) are infinite only at branch-points of the surface, and

* This result implies that the relations are independent of one another, which is the case
in general : but it is conceivable that special relations might exist among the branch-points, which
would affect all these numbers.

t This result applies only to a completely general surface of class p. And, for special forms
of surface of class p, a lower limit for /* can be obtained ; thus, in the case of a two-sheeted
surface, the limit is 2. (See Klein-Fricke, i, p. 556.)

241.] ALGEBRAICAL FUNCTIONS 461

the total number of infinities is 2p — 2 + 2?i. For, let w (z) be the most
general integral of the first kind, and let

Near an ordinary point a on the surface we have

w (z) = iv (a) + (z - a) P (z — a),

where P is a converging series that may, in general, be assumed not to vanish
for z = a ; hence

that is, (j) (z} is finite at an ordinary point.

Near z = oo (supposed not to be a branch-point) we have, if K be the
value of w there,

W-K = -P(-},

z \zj
where P f - j may, in general, be assumed not to vanish for z = oo ; so that

and therefore <£ (z) has a zero of the second order at z = oo .

Near a branch-point 7, where m sheets of the surface are connected, we
have i_ j_

w (z) -w(y) = (z- 7)™ P {(z - 7)™},

where P may, in general, be assumed not to vanish for z = y: hence

_«
</>(*) = (* -7)

so that <f> (z) is infinite at z = 7, and the infinity is of order m—l.

Hence the total number of infinities is 2(w — 1), where m is the number
of sheets connected at a branch-point and the summation extends over all
the r branch-points. But 2p + 1 = S (m — 1) — 2n + 3, and therefore the
number of infinities is 2p — 2 + 2w.

We can now prove that the number of zeros of <f> (z) in the finite part
of the surface is Zp — 2, of ivhich p — 1 can be arbitrarily assigned.

The total number of zeros is 2p — 2 + 2n, being equal to the number of
infinities because </> (z} is an algebraical function. But (f> (z) has been proved
to have a zero of the second order when z = oo and this occurs in each of the
n sheets, so that 2n (and no more) of the infinities of <£ (z) are given by
z = oo . There thus remain 2p — 2 zeros, distributed in the finite part of the
surface.

Moreover, the most general function <£ (z} of the present kind is of the
form

0 (z) = C& (z)

462 ALGEBRAICAL FUNCTIONS [241.

where ^(z), ..., <$>p(z) are derived through the normal functions of the first
kind. The p — ~L ratios of the constants C can be chosen so as to make </> (z)
vanish for p — 1 arbitrarily assigned points. Hence, except as to a constant
factor, an algebraical function arising as the derivative of an integral of the
first kind is determined, save as to a constant factor, by the assignment of p — 1
of its zeros in the finite part of the plane.

Note*. It may happen that the assumptions as to the forms of the
series in the vicinity of a particular point a, of CXD , and of 7 are not justified.

If (f> (a) vanish, we may regard a as one of the 2p — 2 zeros.

If z = GO on one sheet be a zero of <f> (z) of order higher than two, say
2+5, we may consider that s of the 2p — 2 zeros are removed from the finite

part of the surface to coincide with z = oo .

i_

If P {(z — y)m] vanish for z = y, the order of the infinity for <£ (z) is
reduced from m — 1 to, say, m—s—I; we may then consider that s of the
2p — 2 zeros coincide with the branch-point.

242. The existence of functions that are uniform on the surface and,
except at points where they have assigned algebraical infinities, are finite
and continuous, has now been proved ; we proceed, as in § 99, to shew how
algebraical functions imply the existence of a fundamental equation, now to
be associated with the given surface.

The assigned algebraical infinities may be either at the branch-points, or
at ordinary points which are singularities only of the branch associated with
the sheet in which the ordinary points lie, or both at branch-points and
at ordinary points.

Let the surface have n sheets; on the surface let the points. Cj, c.,, ..., cm
be ordinary infinities of orders ql} qy, ..., qm respectively — we shall restrict
ourselves to the more special case in which q1} q2, ..., qm are finite integers,
thus excluding (merely for the present purpose) the case of isolated essential
singularities; and let the branch-points alt a2, ... be of orders pl} p2, ... as
infinitiesf and of orders r, — 1, ra - 1, ... as winding-points.

Let ivl, w.2, ..., wn be the n values of the function for one and the same
algebraical value of z ; and consider the function (w — Wj) (w — w2)... (w - wn).
The coefficients of w are symmetrical functions of the values w1} ..., wn of the
assigned function.

An ordinary point for all the branches w is an ordinary point for each of
the coefficients.

* See Klein-Fricke, vol. i, p. 545.

t A branch-point a is said to be an infinity of order p and a winding-point of order r-1,

.P. I

when the affected branches in its vicinity can be expressed in the form (z - a) r P {(z - a)r}, where

P is finite when z = a.

242.] FUNDAMENTAL EQUATION FOR THE SURFACE 468

An ordinary singularity of order q for any branch, which can occur only
for one branch, is an ordinary singularity of the same order for each of the
symmetric functions ; and therefore, merely on the score of all the ordinary
singularities, each of these symmetric functions can be expressed as a mero-
morphic function the denominator of which is the same rational integral

771

algebraical function of degree 2 qs in z.

s=l

In the vicinity of the branch-point «j , there are r-i branches obtained from

_

(where P is finite when z = a^, by assigning to (z — a^1 its rx various values.
Then, as in § 99, the point ax is no longer a branch-point of any of the
symmetric functions ; and for some of the symmetric functions the point
ttj is an accidental singularity of order p1} but for no one of them is it a
singularity of higher order. Hence, merely on the score of the infinities at
branch-points, each of the symmetric functions can be expressed as a mero-
morphic function the denominator of which is the same rational algebraical
meromorphic function of degree 'Zpl in z.

No other points on the surface need be taken into account. If, then, P (z)
be the denominator of the coefficients arising through the isolated algebraical

in

singularities, so that P (z) is of degree S qs in z, and if Q(z) be the de-

s=l

nominator of the coefficients arising through the infinities at the branch-

)ints, then

P (z) Q (z} (w — Wj) (w — w2) •••(w — wn)

is a rational integral uniform algebraical function of w and z\ say /"(«;, z\

m

fhich is evidently of degree n in w and of degree 2 qg + Sp in z.

s = l

Its only roots are w = wl) ..., wn\ that is, the function w on the Riemann's
surface is determined as the root of the equation f(w, z) = Q] and therefore
the equation f(w, z) = 0 is a fundamental equation, to be associated with
the surface.

Ex. 1. Shew that a fundamental equation for a three-sheeted surface, having <?mm (for
i = 0, 1, ... , 5) for branch-points each of the first order, is

id that a fundamental equation for a four-sheeted surface having the same branch-points
ich of the same order is

(Thomse.)

Every algebraical function on the surface requires its own fundamental
juation ; but, as the branch-points are the same for any surface, no
mdameutal equation can be regarded as unique. Having now obtained

one fundamental equation for algebraical functions on the surface, all the

investigations in chap. XVI. may be applied.

464 APPELL'S FACTORIAL FUNCTIONS [242.

The preceding sketch, in §§ 240 — 242, of algebraical functions is intended only as an
introduction ; the developments are closely connected with the theory of Abelian functions
and of curves. The propositions actually given are based upon

Riemann, Theorie der Abel' schen Function, Ges. Werke, pp. 100 — 102;
Roch, Crelle, t. Ixiv, (1865), pp. 372—376;

Klein's Vorlesunaen ilber die Theorie der elliptischen Modulfunctionen, (Fricke), vol. i,

pp. 540—549 ;

for further information reference should be made to the following sources : —
Brill und Noether, Math. Ann., t. vii, (1874), pp. 269—310;
Lindemann, Untersuchungen iiber den Riemann- Rock' schen Satz, (Leipzig, Teubner,

1879), 40 pp. ;
Brill, Math. Ann., t. xxxi, (1888), pp. 374—409; ib., t. xxxvi, (1890), pp. 321—360.

Ex. 2. Prove that the algebraical equation which subsists (§ 118) between two
functions u and v of a variable z, doubly-periodic in the same periods, is of class either
zero or unity ; that it is of class unity, if only one incongruent value of z correspond to
given values of u and v ; and that it is of class zero, if more than one incongruent value of
z correspond to given values of u and v. (Humbert, Giinther.)

Ex. 3. If between two uniform analytical functions P and Q, which have an isolated
point for their essential singularity, there exist an algebraical relation, then, when either
is regarded as the independent variable, the connectivity of the Riemanri's surface for the
representation of the other is not greater than three. (Picard.)

243. We now pass to the consideration of another class of functions
associated with a Riemann's surface.

The classes of pseudo-periodic functions, which have been discussed,
originally occurred in connection with the functions that are doubly-periodic
functions of the first kind ; and it may, therefore, be expected that, in a
discussion of functions which are multiply-periodic, similar pseudo-periodic
functions will occur.

These functions, in particular such as are the generalisation of doubly-
periodic functions of the second kind, have been considered in great detail by
Appell* ; they may be called factorial functions^.

But the essential difference between the former classes of functions and
the present class is that now the argument of the function is a variable of
position on the Riemann's surface and not, as before, an integral of the first
kind. It is only in subsequent developments of the theory of these functions
that the corresponding modification of argument takes place ; and a factorial
function then becomes a pseudo-periodic function of those integrals of the
first kind.

* " Sur les integrates des fonctions a multiplicateurs... " (Mem. Cour.), Acta Math., t. xiii,
(1890), 174 pp. This volume is prefaced by an interesting report, due to Hermite, on Appell's
memoir. •

They are also discussed in Neumann's Abel' schen Functionen, pp. 273 — 278 ; in Briot's Theorie
des fonctions Abeliennes; in a memoir by Appell, Liouville, 3me Ser., t. ix, pp. 5—24 ; and they
occur in a memoir by Prym, Crelle, t. Ixx, (1869), pp. 354 — 362.

t Fonctions a multiplicateurs, by Appell.

243.]

FACTORIAL FUNCTIONS

465

We consider a Riemann's surface of connectivity 2p + 1, reduced to simple
connectivity by 2p cross-cuts taken, as in § 181, to be a1} blt c2 + a2, 62, ...,
cp + ap, bp. The functions already considered are such that their values
at points on opposite edges of a cross-cut differ by additive constants,
which are integral linear combinations of the cross-cut constants, necessarily
zero for the portions c in the case of all the functions ; the values of the
constants for the cuts a and the cuts 6 depend upon the character of the
functions and are simultaneously zero only when the function is a uniform
function of position on the Riemann's surface, that is, is a rational function of
w and z when the surface is associated with the fundamental equation

F(w, z) = 0.

A factorial function is defined as a uniform function of position on the
resolved Riemann's surface, finite at the branch-points no one of which is at
infinity ; all its infinities are accidental singularities, so that it has no
logarithmic infinities : and at two (practically coincident) points on opposite
edges of a cross-cut the quotient of its values is independent of the point,
being a factor (or multiplier) that is the same along the cut for all parts which
can be reached without crossing another cut.

Then for any portion c the factor is unity, for any cut a it is the same along
its whole length, and for any cut b it is the same along its whole length.

In order to consider the effect of passage over another cross-cut on the con
stant factor, we take the figures of §§ 196,
230. Where ar and br intersect, we have
F(z^ = mr F(ZZ), F(zt) = mr/F(z3) ;

P(z\ — rt'F(^\ W(^\ — n F(r\-

-L \6$) ~~" 'vf J- \^1/J •* \^3/ — fvyJs \-^2/ t

where mr, mr' ; nr, nr' are the constants
for the portions of the cuts ar and 6,..
From these equations it follows that

F (z4) = nrmr'F (z,\ Fig- 85.

and also =nr'mrF(z2),

so that nrmr' = nr'mr.

Again, where cr+l cuts br, we have

F(z5') = nr'F(zs\ F(z6') = nrF(zs),

so that, as F (z5f) = F (z6') when the points are infinitely close together, we
have

or the multiplier lr for cr+1 is lr+1 =

whence
F.

mr

80

466 . FACTORIAL FUNCTIONS IN TERMS OF [243.

Now ttj is met only by 6X and by no cut c : so that TOX = TO/. Hence n^ = w/,
and therefore I2 = l. Hence mz = TO/ ; n2 = n2' and therefore I3 = l; and so on,

so that

lr+i = 1, TO/ = mr, nr' = nr,

the results necessary to establish the proposition.

We shall therefore take the factor along ar to be mr, and the factor along
br to be nr, for r = l, ..., p: and, by reference to § 196, the function at the
positive edge is equal to the function at the negative edge multiplied by the
factor of the cut.

244. Before passing on to obtain expressions for factorial functions in
terms of functions already known, we may shew that all factorial functions
with assigned factors are of the form

<E> 0) R (w, z\

where <I> (z) is a factorial function with the assigned factors and R (w, z) is a
function of w and z, uniform on the Riemann's surface. For if *& (z) and
<& (z) be factorial functions with the same factors, then ^ (/)-=- <!> (z) has its
factors unity at all the cross-cuts, so that it is a uniform function of position
on the surface and is therefore* of the form R (w, z}. It is therefore sufficient
at present to obtain some one factorial function with assigned factors

TOJ, ..., mp, n^, ..., np.

Let Wi(z\ Wz(z\ ..., wp(z) be the p normal functions of the first kind
connected with a Riemann's surface, with their periods as given in § 235.

Let TTi (z), instead of OTJ, of § 237, denote an elementary normal function
of the third kind, having logarithmic infinities at a: and /3j such that, in the
vicinities of these points, the respective expressions for TTI (z) are

- log (z -aJ+P (z - ttj),

and + log (z -&) + Q(z- A) ;

then the period of TTJ (z) for the cross-cut ar is zero, and the period for the
cross-cut br is

for r = I, 2, ..., p. It therefore follows that <J>i (z), where

^ (,) = «*»;

is uniform on the resolved Riemann's surface : it has a single zero (of the first
order) at & and a single accidental singularity (of the first order) at «j ; its
factor for the cross-cut ar is unity and its factor for the cross-cut br is

* It may be pointed out that this result is an illustration of the remark, at the beginning of
§ 243, that the factorial functions have a uniform function of position on the surface for their
argument and not the integrals of the first kind, of which that variable of position is a multiply-
periodic function.

244.] FUNCTIONS OF THE FIRST KIND 467

The function ^>l (z) may therefore be regarded as an element for the repre
sentation of a factorial function.

Let <E> (z) be a factorial function on the Riemann's surface with given
multipliers m and n ; and let it have a number q of zeros &, $>,..., ftq, each
of the first order, and the same number q of simple accidental singularities
«!, a2, ..., aq, each of the first order, and no others. Then $>' (z)/<& (z) has 2q
accidental singularities ; in the vicinity of the q points /3, it is of the form

and in the vicinity of the q points a it is of the form

z — a

cjy (z\ q
hence vjj' _.£«.,(,)

3>' (z)
is finite in the vicinity of all the singularities of , ^ . Thus

log 3> 0) - 2 TT, 0)

S — 1

las no logarithmic infinities on the surface : neither log <£ (z) nor any one
)f the functions ir(z) has infinities of any other kind; and therefore the
foregoing function is finite everywhere on the surface. It is thus an integral
)f the first kind and is expressible in the form

2A1w1 (z) + 2X2w2 (z) + ... + 2\pWp (z) + constant.

2 ws(z)+2 2 \kWk(z)

Hence Q (z) = Aes=l ,

where A is a constant.

The function represented by the right-hand side evidently has the q
points ft as simple zeros and the q points a as simple accidental infinities,
and no others. Higher order of a zero or an infinity is permitted by repeti
tions in the respective assigned series.

In order that it may acquire the factor mr on passing from the negative
. edge to the positive edge of the cross-cut ar, we have

mr = e2Ar7™ ;

and that it may acquire the factor nr in passing from the negative edge to
the positive edge of the cross-cut br, we have

2 2 {Wr (ft) -«"•(«»)} + 2 2 A*£*r

The former equations determine the constants \r in the form

30—2

468 ZEROS AND INFINITIES [244.

for r = 1, 2, . . . , p ; and then the latter equations give

I {wr (&) - wr (a,)} = £log ?ir - 5-. 2 (5fe. log mk\

s=l •67rtfc = l

for r = l, 2, ..., p.

Apparently, \r is determinate save as to an additive integer, say Mr ; and
the value of ^logw,. is determinate save as to an additive quantity, say Nriri,
where Nr is an integer. The left-hand side of the derived set of equations
being definite, these integers Nr and Mr must be subject to the equations

p

iriNr= 2 JffrBir
fc=l

for r = 1, 2, . . . , p ; and therefore, equating the real parts (§ 235), we have

±* p

so that 2, 2 MkMrpkr = 0,

which, by § 235, can be satisfied only if all the integers Mr vanish and there
fore also the integers Nr.

Hence when the foregoing equations connecting the quantities a, 0, log n,
log m are satisfied, as they must be, for one set of values of log n and log m,
that set may be taken as the definite set of values ; and the only way in
which variation can enter is through the multiplicity in value of the functions
w1} ..., wp, which may be supposed definitely assigned.

The expression for the function <&(z) is therefore

- 2 {

the q zeros /3 and the q simple poles a. being subject to the equations
2 {wr(@s)-wr(as)}=%\ognr-z—. 2 (Bkrlogmk).

S = l ^7T1 & = 1

COROLLARY I. The function <&(z) is a rational function of position on
the surface, that is, of w and z, if all the factors n and m be unity. Such a
function has been proved (§ 194) to have as many infinities as zeros; and
therefore integers N^, ..., Np', MI, ..., Mp' exist such that, between the zeros and
the infinities of a rational algebraical function of w and z, the p equations

2 K (&) - wr (a,)} = iriNr - I Mk'Bkr,

s=l k=\

for r = 1, 2, . . . , p, subsist*.

The function 3> (z) then corresponds to a rational algebraical function,
when regarded as a product of simple factors, in the same way as the expres
sion (§ 241) in terms of normal elementary functions of the second kind
corresponds to the function, when regarded as a sum of simple fractions.

* Neumann, p. 275.

244.] OF FACTORIAL FUNCTIONS 469

COROLLARY II. Every factorial function has as many zeros as it has
infinities.

For if a special function <E> (z), with the given factors and possessing q zeros
and q infinities, be formed, every other function with those factors is included
in the form

where R (w, z) is a rational algebraical function of w and z. But R (w, z) has
as many zeros as it has infinities ; and therefore the property holds of F(z).

Further, it is easy to see that the equations of relation between the zeros,
the infinities and the multipliers are satisfied for F(z). For among the zeros
and the infinities of <£ (z), the relations

9 1 P

2, [wr (ftt) - wr («,)} = | log nr - -A— . 2, (Bkr log mk)

k=\

are satisfied ; and among the zeros and the infinities of R (w, z) the relations

2 Wr (&') - Wr («/) = TriNr' - I (Bkr Mk')

are satisfied, where Nr' and the coefficients M ' are integers. Hence, among
the zeros and the infinities of F ' (z), the relations

•o IP

2 {wr (zero) — wr (oo )] = | (log nr + Nr' 2tri) — ^ . z [Bkl, (log mk + 2Mk7ri)}
are satisfied, giving the same multipliers nr and mr as for the special function

COROLLARY III. It is possible to have factorial functions without zeros
and therefore without infinities : but the multipliers cannot be arbitrarily
assigned.

Such a function is evidently given by

lerived from 3?(z) by dropping from the exponential the terms dependent
upon the functions TT(Z). The relations between the factors are easily
obtained.

245. The effect of the p relations

q- 1 p

1 K (&) - wr (as)} = I log nr -^- 2 (Bkr log mk)

subsisting between the factors, the zeros and the infinities of the factorial
function, varies according to the magnitude of q.

If q be equal to or be greater than p, it is evident that all the infinities a
and q— p of the zeros /3 can be assumed at will and that the above relations
determine the p remaining zeros. The function therefore involves 2q - p
arbitrary elements, in addition to the unessential constant A.

470 FUNCTIONS DEFINED BY [245.

In particular, when q is equal to p, the infinities a can be chosen at will
and the zeros ft are then determined by the relations. It therefore appears
that a factorial function, which has only p infinities, is determined by its
infinities and its cross-cut factors.

When q is greater than p, say =p + r, then the q infinities and r zeros
may be chosen at will. By assigning various sets of r zeros with a given set
of infinities, various functions ^ (z), <J>2 (z), . . . will be obtained all having the
same infinities and the same cross-cut factors. Let s such functions have
been obtained ; consider the function

3> 0) = yuA (z) + (*2<£.2 (*)+... + p,®, (z} :

it will evidently have the assigned infinities and the assigned cross-cut
factors. Then s — 1 ratios of the quantities //, can be chosen so as to cause
<& (z) to acquire s — 1 arbitrary zeros. The greatest number of arbitrary
zeros that can be assigned to a function is r, which is therefore the greatest
value of s — 1. Hence it follows that r+1 linearly independent factorial
functions ^ (z), ..., <&r+1 (z) exist having assigned cross-cut factors and p -{-r
assigned infinities ; and every other factorial function with those infinities and
cross-cut factors can be expressed in the form

P&i 0) + /^3>2 (*)+... + /*,.+i<£r+i <»,

where /^ , . . . , /ir+1 are constants whose ratios can be used to assign r arbitrary
zeros to the function.

These factorial functions are used by Appell to construct new classes of functions in a
manner similar to that in which Riemann constructs the Abelian transcendents. Their
properties are developed on the basis of algebraical functions ; but as only the introduction
to the theory can be given here, recourse must be had to AppelPs interesting memoir,
already cited.

246. Various examples of functions defined by differential equations ojR
the first order have occurred, all the equations being of the form

where F is a rational, integral, algebraical function of w and ,- . This is a

CLZ

special form of the more general equation

of the first order : the theorem, that such an equation determines a function,
and the discussion of the characteristics of the function so determined, belong
to the theory of differential equations. In this place we shall consider* the
special form of differential equation, not in its generality but only in the
limited instances in which the function, determined by it, is a uniform function
of z.

* The following investigation has been placed here and not earlier, in order to avoid inter
rupting the development of the preceding theory.

246.] DIFFERENTIAL EQUATIONS OF THE FIRST ORDER 471

Let the equation be of the with degree in -, , supposed irreducible;

when arranged in powers of the derivative, it takes the form
dw\m

)

Because w is a uniform function of z, it has, qua function of z, no branch
points ; and ~r has, qua function of z, no branch-points. Hence infinities of

w are infinities of -, and vice versa ; and therefore -j- cannot become infinite
dz dz

for a finite value of w. It follows that the coefficients /i (w), /2 (w), ... of the
various powers of the derivative are integral functions of w ; they are known,
by the character of the equation, to be rational and algebraical.

Moreover all the general properties possessed by w are possessed by its
reciprocal M = — . When u is made the dependent variable, we have

dz) \dzj J \u
as the equation determining u. Now -7- cannot become infinite except for

CLZ

infinite values of u, for u is a uniform function of z ; hence the coefficients of
powers of -j- must be rational integral algebraical functions of u. This con
dition can be satisfied only if fs (w) be of degree in w not higher than 2s.

Hence, denoting -7- by W and -7- by U, we have the theorem : —
dz (Hz

I. The differential equation

F(W,w)= Wm + F™-1/! (w) + W™~% (w) + . . . = 0
cannot determine w as a uniform function of z, unless the coefficients

/iO), f2(w), f3(w), ...

are rational integral algebraical functions of w of degrees not higher than
2, 4, 6, ... respectively : and when this condition is satisfied, it is satisfied also
for the equation

which determines u, the reciprocal of w.

247. The equation, in the first instance, determines W as a function
of w ; and values of w may be ordinary points or may be branch-points
for W, qua function of w. In the vicinity of such points, it is necessary
to secure that w, as depending upon z, shall be uniform.

472 CRITICAL POINTS OF FUNCTIONS [247.

First, consider finite values for w: let w = 7. For points in the
immediate vicinity of that value, the values of W are not infinite : they
may be

(i) distinct from one another, and no one of them zero at the
point; or

(ii) distinct from one another and at least one of them zero at the
point; or

(iii) not distinct from one another, so that w = 7 is then a branch
point of the function.

(i) Let any value T, a constant different from zero, be the value of
W for w = 7. Then in the vicinity we have

and therefore Tdz =

I + \ (w - 7) + p (w - 7)2 + . . .
= {1 + 2V (w - 7) + 3/ (w - 7)2 + ...} dw,

where X', /u/, . . . are constants. Hence if z0 be the value of z when w = 7,
we have

r (z -z0) = w - 7 + V (w - 7)2 + p (w - 7)3 + ... ,
and the inversion of this equation gives

w - 7 = T (z - z,) + P (z - z,\

that is, w is then a uniform function of z in the vicinity of z0. No new
condition, attaching to the original equation, arises.

(ii) Since the values are distinct from one another, and at least one
of them is zero for w = 7, we must have

— = a(w-y)n{l + b(w-y) + c(w- 7)2 + ...}

for at least one of the values of W, n being an integer. Now as 7 is not a
branch-point, it follows from § 97 that n is equal either to 1 or to 2.
First, if n be unity, we have

so that log (w — 7) + P (w — j) = az,

the constant of integration being absorbed in P (w — 7). Thus

(w-^eP(w~y) = eaz>

and therefore, inverting the functional relation,

that is, w is a uniform function in the vicinity of its own value 7, but it can
acquire this value only for logarithmically infinite values of z. No new
condition, attaching to the original equation, arises.

247.] DEFINED BY DIFFERENTIAL EQUATIONS 473

Secondly, if n be 2, so that

-r- = a (w - y)2 (1 + b (w - y) + c (w - y)- +...},
then, proceeding as before, we have

b log (w — y) + Q (w — 7) = az.

w — y

If b be different from zero, then, as on pp. 474, 475, it can be proved that w
is not uniform in the vicinity of z = oc . Hence b must be zero, so that

i «m

w — v = o — ,

az \azj

giving w as a uniform function of z in the vicinity of its own value y. In
this case w can acquire the value y only for algebraically infinite values of z.
The new condition, attaching to the original equation, will be included in a
subsequent case (III., § 248).

(iii) If w = y be a branch-point, then two cases arise according as W
is not, or is, zero : it cannot be infinite, because y is not infinite.

If W be not zero, we have the value of W in the form

i 2

W=a{I + b (W - y)P + c O - y)P + ...},

where p is a positive integer. The integral of this equation is of the form

1 2

(w - y) {1 + b' (w - y)P + c' (w - y)P + . . .} = a (z - a),

and this makes w uniform in the vicinity of Z = CL, only if powers of w — y
with non-integral indices be absent from the last equation and therefore
also from the former. When the fractional powers are absent from the
former, the implication is that w — y is really not a branch-point for W,
qua function of w, but only that more than one of its values are equal to
a ; then w is a uniform function of z, and therefore W is a uniform function
of w, and vice versa.

If however W be zero at the branch-point, then its value in the
vicinity takes the form

q q+l <?+2

W=a(w-y)v + b(w-y} p +C(W-J)P +...;

and, as W cannot be infinite for a finite value of w, the fraction q/p is
positive. It may be less than 1, equal to 1, or greater than 1. Hence : —

II. If any finite value y of w be a branch-point of W regarded as a
function of w, then, in order that w may be uniform, all the values of W
affected by the point must be zero for w = y.

474 FORM OF FUNCTIONS [248.

248. If q/p < I, the integration of the equation leads to a relation of the
form

p-q p-g + l

Z — 0. = a'(W — y)P + 1)' (W _ ry) P + ......

in which all the indices are positive. The inversion of this relation
makes w uniform in the vicinity of z = a, only if p — q be unity, that is,

if the zero of W as a function of w be of degree 1 — , when the degree is
less than unity ; and the value of z is finite.

If q/p = 1, then we have

i 2

W = a (w — <y) + b(iv — y) p + c(w-j) p + ...

7 12

and therefore a dz = \ 1 + a (w - <V)P + b' (w — y)p + ...}

w — 7

1 2

so that az = log (w - 7) + a" (w — 7)? + b" (w - J)P + ....

az

Let w — y = vp, Z=eP ; then this equation becomes

p log Z = p log v + a"v + b"v2 + ... ,
that is, Z = ve^+*v2+- = vP (v) ;

whence, by inversion, we have a relation of the form

so that tv-<y = eazQ (e?),

shewing that w is uniform for values in the vicinity of w = 7 : it is simply-

periodic in that vicinity, the period being ----- , and it can acquire the value
7 only for (logarithmically) infinite values of z.

If q/p > 1, let q =p + n, where n and p are prime to one another ; then we
have

i+- i+—

W = a (w — 7) P + b (w — 7) P + ...,

so that

-i_- -i-n~^. _i_w^?

adz= {(w-y)~ ~* + &'(*0-7) ~~P+c'(w-y) ~*~+...}dw,

_n n-1

or z = a (w — 7) P + /? (w — 7) P + ...

_l 1

+ B (w - 7) P + € log (w - 7) + P {(to - y)P}.

Hence w can acquire its value 7 only for (algebraically) infinite values of z.

As a first condition for uniformity, the coefficient e must vanish, that is, in

dz ±

the expansion of -^— in powers of (w — 7)?, there must be no term involving

(w - 7)-1. For let

248.] NEAR CRITICAL POINTS 475

so that vn = ZH{a + {3v+ ... + 8vn-1 + evn logv + vnP (v)}.

Then, if v = uZ,

we have un = Q (uZ) + eunZn (log u + log Z),

where Q is a series of integral powers of uZ converging for sufficiently small

values of \uZ\.

Since z is infinitely large for sufficiently small values of w — y, we have
Z infinitesimally small. When Z = 0, the value of Zn log Z is zero ; but for
values of Z that are not zero, the quantity has an infinite number of different
values of the form

Zn (Log Z + 2m7rt),

and there will then be an infinite number of distinct equations determining
u, one corresponding to each of the values of ra. Hence u (and therefore v,
and therefore also w — 7), in that case, has an infinite number of distinct
branches in the vicinity of Z = 0 ; then w is not uniform in the vicinity of
Z = 0. As a first condition for uniformity, we must therefore have e = 0.

We take e = 0 : then the equation between z and v, where w — y = v1', is

z — v~n [a. + (3v + yv" + ...},

the inversion of which can give v (and therefore can give w — 7) as a uniform
function of z, only if n = 1. When n = 1, we have w — y uniform ; and w can
obtain its value 7 only for algebraically infinite values of z.

Combining these results, we have the theorem :

III. If for a finite value y of w, which is a branch-point of W, the
equation in W has a zero for p branches, then, in order that w may be uniform,

the degree of that zero is of one of the forms 1 -- ,1, and 1 + - ; and if it be

of the form* 1 +-, the term in (w — y)-1 must be absent from the expression of

dz .

T— in powers of w — y.

249. Only finite values of w have been considered. For the consideration
of infinite values of w, we pass to the equation in u : and only zero values of
u need be taken into account. If w be uniform, u also is uniform and vice
versa ; hence : —

IV. In order that the function w may be uniform when its value tends to
become infinitely large, the conditions in II. and III. must apply to the equation
in u for the value u = 0.

The branch-points of W, regarded as a function of w, as well as points
where the roots though equal are distinct as in II., are (in addition possibly
to u = 0) the common roots of the equations

The c&sep-l occurs in (ii), § 247 : it will now be included in III.

476 CLASSES OF UNIFORM FUNCTIONS [249.

If, then, the conditions in II. and III. be satisfied for all these points, and if
the conditions in IV. be satisfied for u = Q, that is, for infinite values of w, then
the integral of the equation

. dw

is a uniform function of z.

250. The classes of uniform functions of z can be obtained as follows.
The function, inverse to w, is given by the equation

i-"**-

dw

and therefore z = I ^ .

Let the Riemann's surface for the algebraical equation

regarded as an equation between a dependent variable W and an independent

variable w capable of assuming all values, be constructed ; and let its con-

r rf-7
nectivity be 2P + 1. Then I ^r is the integral of a uniform function of

position on the surface ; and if w0 be a value at any point, then all other
values at that point differ from w0 by integral multiples of

(i) the moduli of the integral at the 2P cross-cuts,

(ii) the moduli of the integral at such other cross-cuts as may be necess
ary on account of the expression of the subject of integration
as a function of w.

Hence the argument of w, a uniform function of z, is of the form z + SmQ,
where the coefficients m are integers and the quantities O are constant.

It has already been proved that uniform functions of z with more than
two linearly independent periods cannot exist ; hence there are at the utmost
two moduli, and therefore, taking account of the results of §§ 235 — 242, it
follows that the uniform function of z is either
(i) a doubly -periodic function of z ; or
(ii) a simply -periodic function of z ; or
(iii) a rational function of z.

Further*, the class of the Riemanns surface for the equation f(W, w) = 0 is
either unity or zero ; for in what precedes, the value of P is not greater than
unity, when the limitations as to the possible number of periods are assigned.

It is now easy to assign the criteria determining the class of functions to

* This result is due to Hermite, and is stated by him in a letter to Cayley, Land. Math. Soc.,
t. iv, (1873), pp. 343 — 345. The limitation of the class to zero or unity is not, in itself, sufficient
to ensure that w is a uniform function of z.

250.] DEFINED BY DIFFERENTIAL EQUATIONS 477

which w belongs, when it is known to be a uniform function of z satisfying the
differential equation.

(i) Let w be a uniform doubly-periodic function. Take any parallelogram
of periods in the finite part of the plane : all values of z within the parallelo
gram are finite, and all possible values of w are acquired within the parallelo
gram.

Let 7 be a finite value of w for a point z — c ; then, since the function is

uniform, we have

w - 7 = (z - c}m P (z - c),

where m is an integer and P(z — c) does not vanish for z = c : and, by inversion,
we also have

where Q is finite but does not vanish for w = 7.

Now d~ = (z- c)"1-1 [mP (z - c) -f (z - c) P' (z - c)}

dz

where Ql does not vanish for w = 7.

If m = 1, then 7 is an ordinary point for —, .

CLZ

If m > 1, then 7 is a zero branch-point for W, of index-degree equal to

i-l.

m

If, in the vicinity of z = b, w be infinitely large of order q, then z = b is a
zero of u of order q, so that we have

• -(*-t)fPa(jr-ft);

as in the first of these cases, it follows that

where P2 does not vanish for u — 0.

Hence it follows that if, for finite or for infinite values of w, all the branch
points for W be zeros and each of them have its degree less than unity, the

index of the degree being of the form 1 — , where p is the number of
branches of W affected, then the uniform function w is doubly-periodic.

(ii) Let w be a uniform simply- periodic function, of period w ; then it is
known (§ 113) that w can be expressed in the form

Take any strip in the 2-plane as for a simply-periodic function, bounded by

478 CLASSES OF UNIFORM FUNCTIONS [250.

lines whose inclination to the axis of real quantity is \TT + arg. &>, as in § 111 :
in this strip the function acquires all its values. The variable Z is finite in
the strip except at the infinite limits ; at one infinite limit we have z = kiw,
where k is positive and infinitely great, and then Z = e~^k — 0, and at the
other we can take z = — kiw and then Z = e2wk = oo ; so that Z=Q and oo at
the infinite limits.

2TTCJ

Let 7 be a finite value of w for a finite point z = c and let C = e M : then
we have

w-7=f(Z)-f(C)

= (Z-C)ig(Z-C),

where g (Z — C) does not vanish for Z = C and q is a positive integer.
When q = 1, we have

Z-C = (w-y)G(iu-v),
where G does not vanish for w = 7 ; and then

= H(w-y),
where H does not vanish for w = 7 ; the point w = y is an ordinary

. . dw

point tor -=- .
dz

When q > 1, we have

i i

Z-C = (w-y)v G{(w-yyi},
where G does not vanish for w = 7 ; and then

where h does not vanish for w = 7. Such a point is a branch-zero for q branches
of W, and its index-degree is 1 -- .

If the value of w be infinite for the finite point z = c, then we have

u = (Z-C)<ig(Z-C).

If q = 1, the point is an ordinary point for -7- ; if q > 1, it is a branch-zero

for q branches of -7- and its index-degree is 1 -- .
dz q

When z = oo , then Z = 0 or Z = oo . The value of the function w for
infinite values of z is either finite or infinite.

Let w be a finite quantity 7, for infinitely large values of z. When Z is
very small, we have

250.] DEFINED BY DIFFERENTIAL EQUATIONS 479

where q is a positive integer and /does not vanish for Z = 0 ; and then

j. i

Z=(w-y)ig{(w-y)i],

where g does not vanish for w = y. Then

= Zili (Z\
where h does not vanish when Z = 0 ; and therefore

or the point w = 7 is a branch-zero of q branches of -=- and its index-degree

CbZ

is unity. And when Z is very large, we have

where g is a positive integer and / is finite and not zero for Z = oo . As
before, it is easy to see that

-£=(w-y)Pt{(w-yF},

or the point w = y is a branch-zero of q branches of -,- and its index-degree
is unity.

If, however, the value of w be infinite for infinitely large values of z, then
we have

u = Z<ifl(Z)

when Z is very small, and u = Z~vf2 ( - }

\ZJ

when Z. is very large. As before, the point u = 0 is then, in each case, a
branch- zero of q branches -=- , and its index-degree is unity.

Hence it follows that if all the branch-points of W be zeros, if one of them
have its degree equal to unity, and if all the other branch-zeros are of index-
degree less than unity, the index of the degree being of the form 1 - - ,

where p is the number of branches of W affected, then the uniform function
w determined by the equation /(Tf, w) = 0 is simply-periodic.

(iii) Let w be a rational function of z\ then it can be expressed
in the form

wJW

/,(*)'

where/j and/2 are rational, integral functions of z.

480 ORDERS OF ZEROS [250.

Finite values of w can arise from values of z in the vicinity of (a) a zero of
fi (z), say z = c, or (b) an infinity of f» (z}. For the former, we have, if 7 denote
the value of z,

where F does not vanish for z — c : and then, inverting the functional

relation,

i

z — C = (w — <y)m P(w — 7),
where m is a positive integer which may be 1 or greater than 1.

Now dfa=(z- c">m~l \mF(z -c) + (z-c) F' (z - c)},

so that, if m — 1, we have -5- = Q (w — 7),

dz

where Q does not vanish when w = y; and, if m > 1, we have

-i i

— - — = (w — 7) m Qi \(w — 7) jj
dz

where Q1 does not vanish when ^ = 7. Hence ^ = 7 is either an ordinary
point for W or a branch-point at which m branches vanish, the index-
degree of the zero being 1 .

For an infinity of f2 (z) we must have z = x> ; and therefore, for
infinitely large values of z, we have

\z
where F does not vanish when z = oo . Proceeding as before, we have

where Fl does not vanish when w = y. If X = 1, w — 7 is an ordinary point,
a case which has been considered; if \>l, w = j is a branch-point for

W, at which X branches vanish, and the index-degree of the zero is 1 + - .

A,

Infinite values of w can arise from values of z that are infinitely
Iarg0 — in connection with f-^ (z) — or from values of z that are zeros of the
denominator. For the former, we have

where X is a positive integer and F does not vanish for z = oo ; and then
proceeding as before, we have

11

l + -,

-j-
dz

250.] SUMMARY OF RESULTS 481

so that, if \ = 1, u = Q is an ordinary point, a case of which account has
already been taken ; and if \>l, u = 0 (that is, w = oo ) is a branch-point

for U at which X branches vanish, and the index-degree of the zero is 1 + - .

Moreover, as w is a rational function, we do not have both w = 7 and u = 0
for infinite values of z, unless (possibly) z—x is an essential singularity of
the function.

It thus appears that, when w is a rational algebraical function, there is
only one value of w which, being a branch-point for W, gives m branches

vanishing, the index of the degree of the zero being 1 4- — ; all other branch-

lib

points of W give zeros that are of degree-index less than unity, each being of

the form 1 — , where n is the number of branches that vanish at the point.
n

251. The following is a summary of the results that have been ob
tained : —

I. In order that an irreducible differential equation of the first order
may have a uniform function for its integral, it must be of the form

((Jin

where /j(w), f2(w), ...,/m(w) are rational, integral, algebraical functions of w
of degrees not higher than 2, 4, 6, ..., 2m respectively: and this condition as
to degree is then satisfied for the equation

du\m fdu\m~l , /l\ . /l\

TT) ~ b~ </U-) + ..-± **/*{- 1=0.

dz] \dzj \u) \u)

II. If any finite value of w be a branch-point of W when regarded
as a function of w determined by the equation F(W, w) = 0, then all the
affected values of W must be zero for that value of w ; and likewise for the
value u = 0 in connection with the equation

G(U,u) = 0.

III. If for a value of w, which is a branch-point of W when regarded as
a function of w, there be a multiple root of F( W, w} = 0 which is zero for n
branches, the index-degree for each of those branches is of one of the forms

1 — , 1, 1+-; and likewise for the value w = 0 in connection with the

1\i 71

equation G(U)u) = Q.

IV. The class of the equation F(W, w) = 0, and therefore the class of
the Riemann's surface associated with the equation, is either zero or unity.

F. 31

482 SUMMARY OF RESULTS [251.

V. If all the multiple zero-roots of W, for finite values or for an
infinite value of w, be of index-degree less than unity, each of them being of

the form 1 — , then w is a uniform doubly-periodic function of z.

n

VI. If, for some value of w, there be a single set of m multiple zero-
roots of index-degree equal to unity, and if, for finite values or for an
infinite value of w, all the other sets of multiple zero-roots have their

respective index-degrees less than unity and of the form 1 — , then w is
a uniform singly-periodic function of z.

VII. If, for some value of w, there be a single set of m multiple zero-
roots the index-degree of which is equal to 1 + — , and if, for other values
of w, all the other sets of multiple zero-roots have their respective index-
degrees less than unity and of the forms 1 - - , then w is a rational

algebraical function of z.

In all other cases the equation, supposed irreducible, cannot have a
uniform function of z for its integral. If the equation have a uniform
function of z for its integral, and the preceding conditions in V., VI. or VII.,
be not satisfied, the equation is reducible*, that is, it can be replaced by
rational equations of lower degree to which the criteria apply.

Note. The preceding method may be considered as essentially due to Briot and
Bouquet.

There is another method of proceeding, which leads to the same result. It is based
upon Hermite's theorem (§ 250), proved independently ; and its development will be found
in memoirs by Fuchsf and Eaft'yt A reference to the memoirs which have been quoted
shews that the equation F(W,w) = Q, when it is satisfied by a uniform function of z, can be
associated with the theory of unicursal curves and of bicursal curves.

252. The preceding general results will now be applied to the particular

equation

(dw\s ,, ,

U) =f(w)-

where / is a rational, integral, algebraical function of degree not greater,
than 2s.

Let f(w) = \s(w-a)l(w-b)m...,

* This investigation is based upon two memoirs by Briot et Bouquet, Journ. de VEc. Poly-
technique, t. xxi, Cah. xxxvi, (1856), pp. 134—198, 199—254 ; and upon their Traite des /auctions
elliptiques, pp. 341—350, 376—392. A memoir by Cayley, Proc. Land. Math. Soc., vol. xviii,
(1887), pp. 314—324, may also be consulted.

+ Comptes Eendus, t. xciii, (1881), pp. 1063—1065 ; Sitzungsber. d. Akad. d. Wiss. zu Berlin,
1884, (ii), pp. 709, 710.

J Annales de VEc. Norm., 2™ Ser., t. xii, (1883), pp. 105—190; ib., 3me S&., t. ii, (1885),
pp. 99—112.

252.] BINOMIAL DIFFERENTIAL EQUATIONS 483

where X, a, b, ... are constants and I, m,... are integers, and

The equation in u ( = - ) and -,- is
\ wj dz

thus the values of -,- and -5- are respectively

dw l~ ™

_ = A,(w_a)S(w_6)S _f

du 2-1-™- I ™

~~dz=Xu " S (l~au)s(l-bu)s....

Because the integral of the equation must be uniform, each of the indices
9 I m I m i i

z ~ ~ ~ 7 - •••> ~» — » ••• must be of one of the forms 1 - - , 1, or 1 + - •

P P

and p may be 1, but the point is then not a branch-point. Then the
smallest value of p is 2 and the least index is therefore 1 ; hence, as

I m

- + - + ...<$

s s

there cannot be more than four distinct (that is, non-repeated) factors in/(w).
Hence

(a) if one of the indices -,..., be greater than 1, each of the

o 6

other indices must be less than 1, unless it be 2 when all
the others are zero;

(6) if one of the indices -, -,..., be equal to 1, then either each of

o S

the other indices must be less than 1, or one other is equal to
1, and then there is no remaining index ;

(c) if each of the indices -,—,..., be less than 1, then 2 - - - - - . .

S S 59

may be less than 1, or equal to 1, or greater than 1.

These cases, associated with the possible numbers of factors, will be taken
in order.

I. Let there be a single factor ; the equation is

and therefore (- ~Y = \*u*-i (\ _ auytm

Now -, not being 2, is either 1-1, 1, i + L and these forms cover

" S

also the possible forms of 2 — -

s

31—2

484 CHARACTER OF FUNCTIONS DEFINED [252.

If I = s - 1, then one index (for w = a) is equal to 1 , and the other

S

(for u = 0) is equal to 1 + - : the function w is rational and algebraical in z,

S

and z is infinite only when w = oo : hence the integral w is a rational, integral,
algebraical function of z.

If l = s + I, the reasoning is similar; and the integral is a rational,
algebraical, meromorphic function of z.

If l = s, the indices are each equal to unity: the integral is a simply-
periodic function of z. The equation is reducible.

If I = 2s, the equation is reducible ; the integral is algebraical.

The equations in the respective cases are

•V - / \. n / A \

= \s(w-a)s 1 (A.),

= \s(w- a)s+l (A.),

dw . v, , . N

-y- = \ (w — a)* .............................. (A.),

az

where (A.) implies that the uniform integral is an algebraical function of z,
and (S. P.) implies that it is a simply-periodic function ; the letters (D. P.)
will be used to imply that the uniform integral is a doubly-periodic function.

II. Let there be two distinct factors ; then the equation is

Z] = \°(w-a)l(w-bym.

dzj

First, let one of the indices in the expression for — be greater than 1, say -j
It is not necessarily in its lowest terms ; when reduced to its lowest terms, let

Vft

Then — must be less than 1 ; when reduced to its lowest terms, let
s

m _ i 1

7= ".J1

/ 77?

which is the necessary form. And 2 — _ -- ... must be less than 1, and it

s s

must be expressible in the form 1 -- : hence

2

2~

252.] BY BINOMIAL DIFFERENTIAL EQUATIONS 485

111

and therefore 1 + - = - + - ,

p a T

where p and cr are each greater than unity. If T> 1, the right-hand side is
manifestly less than the left ; and therefore we must have T = 1, p — & ; and
the common value of p and a is s. The integral is then a rational algebraical
function of z.

Secondly, let one of the indices in the expression for -7- be equal to 1 , say

in 1

I = s. Then — is either 1 or of the form 1 .

s cr

If — = 1, the exponent of u in the expression for -=- is zero : the equa
tion is

<dw

,dz
which is reducible ; it has a simply-periodic function for its integral.

If — = 1 , the exponent of u in the expression for -,- is - . This must

s cr dz a

be of the form 1 , so that

P

a- p~
hence, as cr and p are each greater than 1, each must be 2. The equation is

/ckAs

\dz)
which is reducible ; and the integral is a simply-periodic function.

Thirdly, let each of the indices in the expression for -p be less than 1 ; as

they are not necessarily in their lowest terms, let - = 1 , — = 1 . Then

s p s <r

the index of u in the expression for -=- is — h — ; because p and a are each

dz p cr

greater than 1, this index cannot be greater than 1.

If - + - = 1, the only possible values are p = 2, a = 2 ; the equation is
P °~

which is reducible ; the integral is a simply-periodic function of z.

If - + - be less than 1, then, as it is the index of u in the expression for

P a
j- , it must be of the form 1 , where T is greater than 1 : thus

111

- + -+- = 1,
p a- r

486 CHARACTER OF FUNCTIONS DEFINED [252.

„ dw -. du ,11

and then all the indices in the expressions for -j- and -7- are less than 1.

Hence for such equations as exist, the integrals will be doubly-periodic
functions.

In this equation the interchange of p and a- gives no essentially new
arrangement. We must have r > 1 : the solutions for values of r greater
than 1 are : —

(a) T = 2 ; then - + - = 5 , so that p = 3, cr = 6 ; p = 4, <r=4.
p cr L

112

(1) T = 3 ; then - + - = ~ , so that p = 2, o- = 6 ; p = 3, <r = 3.
p cr o

113

(c) T = 4 ; then - + - = T , so that p = 2, <r = 4.
' p cr 4

(c£) T = 5 gives no solution.

(e) T = 6 ; then - + -=-, so that p = 2, <r = 3.
p cr O

And no higher value of T gives solutions.

Hence the whole system of equations, satisfied by a uniform function
of z and having two distinct factors in f(w), is : —

dz
\dz

dz

252.] BY BINOMIAL DIFFERENTIAL EQUATIONS 487

III. Let there be three distinct factors : then the equation is

fdw\s

[T- = V (w - a)1 (w - b}m (w - c}n,

\dz J

and therefore

du>\" _ 8 n,_l_m_nn vn
dz j

If one of the indices in the expression for -»- be greater than 1 sav -= 1 4- -

dz J s p '

then — , - must be of the form 1 , 1 — , where a and r are each greater

S S (T T

than 1.

The index of u in the expression for ~ is then - H 1, a quantity

dz a r p J

which is necessarily negative, for p is finite; and the index should either

be zero or be of a form 1 . Hence no one of the indices - — - can be

H> s s s

greater than 1.

Secondly, let one of the indices in the expression for — be equal to 1, say
I = s. Then since m + n^s, only one of the indices is unity ; and therefore
— , - are of the form 1 , 1 , where p and cr are each greater than 1.

p o p (7

The index of u in the expression for -j- is then - H 1, and it cannot be

dz pa

negative ; hence the only possible values are p = 2 = a, and they make the
index zero. There is thus one index equal to 1, and the others are less than
1 : the integral of the equation is a simply-periodic function of z.

Thirdly, let all the indices in the expression for ~ be less than 1 • then

dz

they are of the forms l--,l--,l_ where p, <r, T are greater than 1 ;

p cr T

and the index of u in the expression for ~ is - H 1 1. Because the

dz par

smallest value of p, a, r is 2, this last index is not greater than £ ; hence it
must be 1 , where, because this quantity is the index of u, //, is equal to 1

or to 2. In either case, all the indices are less than 1 ; and therefore the
integrals of the corresponding equations are doubly-periodic functions of z.

If - + 1 J. 1 _ I » I _ I

p a- T 2'

1113

so that - -\ f- - = - , the only possible solution is

p <T T Z*

P,V,T = 2, 2, 2.

488 FUNCTIONS DEFINED BY [252.

If - H --- 1- - = 1 the only possible solutions are
par

p,a, r = 2, 3, 6;
2,4,4;
3, 3, 3.

Hence the whole system of equations, satisfied by a uniform function of z and
having three distinct factors inf(w), is: —

Y = x* (w - a)2 (w - b) (iv -c) ......... (S.P.),

\dz J

-V(y-a) (w-6) (w-c) ......... (D.P.),(7),

= x« (w - a)3 (w - I)* (w - c)5 ......... (D. P.), (8),

= x4 O - a)2 (w - 6)3 (w - c)3 ......... (D. P.), (9),

(fjin\ 3
^) = X3 (w - a)2 (w - 6)2 (w-c)2 ......... (D. P.), (10).
dz 1

IV. Let there be four distinct factors ; then the equation is
? = v (w - a)1 (w - 6)m (w - c)w (w - d)p.

Since m - , ^ are each of a form 1 - - , and their sum is not greater than
s s s s p

I m n p 1

2, it is easy to see that the only possible solution is given by~ = ~==^ = ^ = 2'

each index is less than 1, and the integral is a doubly -periodic function.

Hence the single equation, satisfied by a uniform function of z and having
four distinct factors in/(w), is

Y= V(w-a) (w-b) (w-c) (w-d) ......... (D.P.), (11).

Those of the complete system of equations, which have their integrals either rational
algebraic functions or simply-periodic functions of 2, are easily integrated. The remainder,
which have uniform doubly-periodic functions of z for their integrals, are most easily
integrated by first determining the irreducible infinities of the functions and their orders :
and then, by the results of Chapters X. and XL, the integral can be constructed.

The irreducible infinities can be determined as follows. In the equation for -^ , let the

index of u be 1 - - ; and let s=o-p. Then the equation which determines u is
P

252.] BINOMIAL DIFFERENTIAL EQUATIONS 489

so that for very small values of u, we have

\u P+...f du—-a\dz,

where a is a primitive sth root of unity. Hence

i
a\ (z-c) = pu>>+...,

and therefore - = u = ap\p(z-c)p + ... .

w

It thus appears that the accidental singularity of w at z = c is of order p ; and, as there
are a- distinct values of ap, there are a- distinct accidental singularities to be associated
with the respective values.

Applying these to the equations which, having doubly-periodic functions for the
integrals, are numbered (1) to (11), we have the following results, where a- is the number
of distinct irreducible accidental singularities and p is the order of each of these
singularities :

number of equation

(1)

(2)

(3)

(4)

(5)

(6)
1

(7)

(8)

(9)

(10)

(11)

number of singularities =<r

3

2

2

1

1

1

2

6

4

3

2

order of singularity = p

2

2

3

3

4

6

1

1

1

1

All the binomial equations, which have uniform functions of z for their integrals, have
been obtained. The general results, summarised in § 251, can be applied to other equa
tions ; the application to trinomial equations will be found in the treatise by Briot and
Bouquet (cited p. 482, note).

Note. The binomial equations can be treated otherwise, by forming the equation

z-a = \(w-a) s(w-b)

but, as indicated at the beginning of § 252, the method in the text is adopted in order to
illustrate the general results of § 251. (See also Note, § 251.)

Ex. 1. Prove that the integral of the equation

^dz) \dz)
is a rational function of z; that the integral of

(dw\3 /c??#\2 2
dzj \dz)

is a simply-periodic function of z ; and that the integral of

f?Y+3(^?Y+*-4«0

is a doubly-periodic function of z.

Find the infinities of each of the functions : and integrate the equations.

(Briot et Bouquet.)

490 EXAMPLES [252.

Ex. 2. Shew that, if an irreducible trinomial equation of the form
(dw\m /rfiA"*-1, , , , , , . .

(is) +(dZ) /I(">+/»M=°

have a uniform integral, then m may not be greater than 5 ; and that, if m be 4 or 5,
the uniform integral is a doubly-periodic function.

Apply this result to the discussion of the equation

fdw\> fdw\* . , _. 4* 9 .
(dz) +(d-Z) 0— l)-5^2(^2-l)4 = 0-

(Briot et Bouquet.)
Ex. 3. Shew that the integral of the equation

/dw\ °

f !j?J = A (w - a)2 (w - &)<•• (y> - c)fi

is a two-valued doubly-periodic function of z. (Schwarz.)

Ex. 4. Shew that, if a function w be determined by a differential equation

where F is a rational integral algebraical function of w and -=- , of degree m in ->- , and

does not contain z explicitly, then to each value of w there correspond m series of values
of z, the terms in each series differing from one another by multiples of periods.

Prove further that, if the integral w have only a limited number of values for each value
of 2, then it is determined by an algebraical relation between w and u, where u may be z,

or e <° , or g>(z). (Briot et Bouquet.)

These results should be compared with the results obtained in Chapter XIII. relative
to functions which possess an algebraical addition -theorem.

CHAPTER XIX.

CONFORMAL REPRESENTATION: INTRODUCTORY.

253. IN § 9 it was proved that a functional relation between two
complex variables w and z can be represented geometrically as a copy of part
of the £-plane made on part of the w-plane. At various stages in the theory
of functions, particularly in connection with their developments in the
vicinity of critical points, considerable use has been made of the geometrical
representation of the analytical relation ; but it has been used in such a way
that, when the equations of transformation define multiform functions, the
branches of the function used are uniform in the represented areas.

The characteristic property of the copy is that angles are preserved, and
that no change is made in the relative positions and (save as to a uniform
magnification) no change is made in the relative distances of points that lie
in the immediate vicinity of a given point in the ^-plane. The leading
feature of this property is maintained over the whole copy for every small
element of area : but the magnification, which is uniform for each element,
is not uniform over the whole of the copy.

Two planes or parts of two planes, thus related, have been said to be
conformally represented, each upon the other.

Now conformal representation of this character is essential to the con
stitution of a geographical map, made as perfect as possible : and a question
is thus suggested whether the foregoing functional relation is substantially
the only form that leads to what may be called geographical similarity. In
this form, the question raises a converse more general than is implied by the
converse of the functional relation, inasmuch as it implies the possibility that
the property can be associated with curved surfaces and not merely with
planes. But a little consideration will shew that the generalisation is a
priori not unjustifiable, because, except at singular points, the elements of the
curved surface can, in this regard, be treated as elements of successive
planes. We therefore have* to determine the most general form of analytical
relation between parts of two surfaces which establishes the property of
conformal similarity between the elements of the surfaces.

* The following investigation is due to Gauss : for references, see p. 500, note.

492 GENERAL PROBLEM [253.

Let x, y, z be the coordinates of a point R of one surface with t, u for its
parameters, so that x, y, z can be expressed in terms of t, u ; and let X, Y, Z
be the coordinates of an associated point R' of the other surface with T, U
for its parameters, so that X, Y, Z can be expressed in terms of T, U. Then
the analytical problem presented is the determination of the most general
relations which, by expressing T and U in terms of t and u, establish the
conformal similarity of the surfaces.

Suppose that G and H are any points on the first surface in the imme
diate proximity of R, and that 0' and H' are the corresponding points on the
second surface in the immediate proximity of R' : then the conformal
similarity requires, and is established by, the conditions : (i), that the ratio
of an arc RG to the corresponding arc R'G' is the same for all infinitesimal
arcs conterminous in R and R respectively; and, (ii), that the inclination of
any two directions RG and RH is the same as the inclination of the cor
responding directions R'G' and R'H'. Let the coordinates of G and of H
relative to R be dx, dy, dz and Bx, By, Sz respectively ; and those of G' and
of H' relative to R be dX, dY, dZ and SX, BY, BZ respectively. Let ds
denote the length of RG and dS that of R'G'; let m be the magnification of
ds into dS, so that

dS = mds,

a relation which holds for every corresponding pair of infinitesimal arcs
at R and R'.

By the expressions of x, y, z in terms of t and u, we have equations of
the form

dx = adt + a'du, dy = bdt + b'du, dz = cdt + c'du,

where the quantities a, b, c, a', b', c' are finite. Let there be some relations,
which must evidently be equivalent to two independent algebraical equations,
expressing T and U as functions of t and u ; then we have equations of the
form

dX = A dt + A'du, dY = Bdt + B'du, dZ = Cdt + C'du,

where the quantities A, B, C, A', B', C' are finite and are dependent partly
upon the known equations of the surface and partly upon the unknown
equations of relation between T, U and t, u. Then

ds2 = (a2 + b2 + c2) dt2 + 2 (aaf + bb' + cc') dtdu + (a2 + b'2 + c'2) du2,
dS2 = (A2 + B2 + C2) dt2 + 2 (AA1 + BB' + CC') dtdu + (A'* + B'2 + C'2) du2.

Since the magnification is to be the same for all corresponding arcs, it
must be independent of particular relations between dt and du ; and there
fore

a2 + 62 + c2 aa' + bb' + cc'
each of these fractions being equal to m2.

253.] OF CONFORMAL REPRESENTATION

Again, since the inclinations of the two directions RG, RH '; and RG',
RH' ; are given by

ds Ss cos GRH

= (a2 + b'2 + c2) dt St + (aa' + W + cc') (dt Su -f St du) + (a 2 + b''2 + c'2) du Su,
dSSS cos G'R'H'

=(A*+B2+C*)dtSt+(AA'+BB'+CC')(dtSii+Stdii)+(A'2+B'-2+C'-2)duSu,
we have, in consequence of the preceding relations,

m?ds Ss cos GRH = dS SS cos G'R'H'.

But dS=mds, SS=mSs; and therefore the angle GRH is equal to the
angle G'R'H'. It thus appears that the two conditions, which make the
magnification at R the same in all directions, are sufficient to make the
inclinations of corresponding arcs the same ; and therefore they are two
equations to determine relations which establish the conformal similarity
of the two surfaces.

These two equations are the conditions that the ratio dS/ds may be
independent of relations between dt and da; it is therefore sufficient, for
the present purpose, to assign the conditions that dS/ds be independent of
values (or the ratio) of differential elements dt and du.

Now ds2 is essentially positive and it is a real quadratic homogeneous
function of these elements ; hence, when resolved into factors linear in the
differential elements, it takes the form

ds* = n (dp + idq) (dp — idq),

where n is a finite and real function of t and u, and dp, dq are real linear
combinations of dt and du. Similarly, we have

d& = N(dP + idQ) (dP - idQ),

where, again, N is a finite and real function of t and u or of T and U, and
dP, dQ are real linear combinations of dt and du or of dT and dU. Thus

_N(dP+idQ)(dP-idQ)

in- — —

n (dp + idq) (dp — idq)

It has been seen that the value of m is to be independent of the values and
of the ratio of the differential elements.

Now taking

aa' + W + cc' q/2 + 6>2 + c/2

so that 0 and </> are, by the two equations of condition, the same for ds and
dS, and denoting by i/r the real quantity (<f> - 02)^, we have

ds2 = (a2 +b2 + c2) {dt + du (6 + 1»} {dt + du (6 - ty)},
and dS* =(A* + &+ C*) {dt + du (0 + i»} {dt + du(8- 1»].

494 GAUSS'S SOLUTION [253.

Then, except as to factors which do not involve infinitesimals, the factors of
ds* and of dS2 are the same. Hence, except as to the former factors, the
numerator of the fraction for m2 is, qua function of the infinitesimal
elements, substantially the same as the denominator; and therefore either

dP + idQ dP-idQ , . . . .

(a) — rr-5 and —, - r-r-^ are finite quantities simultaneously ;

dp + idq dp — idq

or

dP + idQ dP-idQ fi . .... . ,

(p) -j - -_,- and -j - r-j- are finite quantities simultaneously.

Either of these pairs of conditions ensures the required form of m, and so
ensures the conformal similarity of the surfaces.

Ex. Shew that both p and q satisfy the partial differential equation

' '

Consider (a) first. Since (dP + idQ)f(dp + idq) is a finite quantity, the
differentials dP + idQ and dp + idq vanish together and therefore the quan
tities P + iQ and p + iq are constant together. Now P and Q are functions
of the variables which enter into the expressions for p and q ; hence P + iQ
and p + iq, in themselves variable quantities, can be constant together only if

where/ denotes some functional form. This equation implies two independent
relations, because the real parts, and the coefficients of the imaginary parts,
on the two sides of the equation must separately be equal to one another ;
and from these two relations we infer that

P-iQ=f1(p-iq),

where / (p — iq) is the function which results from changing i into — i
throughout f(p + iq) and is equal to f(p — iq), if i enter into /only through
its occurrence in p + iq. From this equation, it follows that

dP-idQ

dp — idq

is finite, and therefore a necessary and sufficient condition for the satisfaction
of (a) is that P, Q and p, q be connected by an equation of the form

P+iQ=f(p + iq).

Moreover, the function / is arbitrary so far as required by the preceding
analysis ; and so the conditions will be satisfied, either if special forms of/ be
assumed or if other (not inconsistent) conditions be assigned so as to deter
mine the form of the function.

Next, consider (/3). We easily see that similar reasoning leads to the
conclusion that the conditions are satisfied, when P, Q and p, q are connected
by an equation of the form

iQ=g(p-iq);

253.] PLANE AS SURFACE OF REFERENCE 495

and similar inferences as to the use of the undetermined functional form of g
may be drawn. Hence we have the theorem : —

Parts of two surfaces may be made to correspond, point by point, in such
a way that their elements are similar to one another, by assigning any
relation between their parameters, of either of the forms

and every such correspondence between two given surfaces is obtained by the
assignment of the proper functional form in one or other of these equations.

254. Suppose now that there is a third surface, any point on which
is determined by parameters A, and /j, ; then it will have conformal similarity
to the first surface, if there be any functional relation of the form

A, + i/j, = h (p + iq).

But if h~l be the inverse of the function h, then we have a relation

P *

which is the necessary and sufficient condition for the conformal similarity
of the second and the third surfaces.

This similarity to one another of two surfaces, each of which can be made
to correspond to a third surface so as to be conformally similar to it, is an
immediate inference from the geometry. It has an important bearing, in the
following manner. If the third surface be one of simple form, so that its
parameters are easily obtainable, there will be a convenience in making it
correspond to one of the first two surfaces so as to have conformal similarity,
and then in making the second of the given surfaces correspond, in conformal
similarity, to the third surface which has already been made conformally
similar to the first of them.

Now the simplest of all surfaces, from the point of view of parametric
expression of points lying on it, is the plane : the parameters are taken to
be the Cartesian coordinates of the point. Hence, in order to map out two
surfaces so that they may be conformally similar, it is sufficient to map
out a plane in conformal similarity to one of them and then to map out
the other in conformal similarity to the mapped plane: that is to say, we
may, without loss of generality, make one of the surfaces a plane, and all
that is then necessary is the determination of a law of conformation.

We therefore take P = X, Q = Y, N= 1 : and then

where Z is the complex variable of a point in the plane ; and the equations
which establish the conformation of the surface with the plane are

=f

496 CONFORMAL REPRESENTATION [254.

where /j (p — iq) is the form of f(p + iq) when, in the latter, the sign of i is
changed throughout.

As yet, only the form P + iQ =f(p + iq) has been taken into account.
It is sufficient for our present purpose, in regard to the alternative form
P + iQ = g (p — iq), to note that, by the introduction of a plane as an inter
mediate surface, there is no essential distinction between the cases*. For
as P = X, Q=Y, we have

X + iY = g(p-iq\

and therefore X — iY— g1 (p + iq),

which maps out the surface on the plane in a copy differing from the copy
determined by X + iY=gl(p + iq)

only in being a reflexion of that former copy in the axis of X. It is therefore
sufficient to consider only the general relation

X + iY=f(p + iq).

Ex. We have an immediate proof that the form of relation between two planes, as
considered in § 9, is the most general form possible. For in the case in which the
second surface is a plane, we have dsi = dx^-if-dyz, so that n=l, p = x, y=--y: hence the
most general law is X+iY=f(x + iy\

that is, w=f(z)

in the earlier notation. Some illustrations arising out of particular forms of the function
/ will be considered later (§ 257).

255. In the case of a surface of revolution, it is convenient to take <£ as
the orientation of a meridian through any point, that is, the longitude of the
point, or as the distance along the meridian from the pole, and q as the
perpendicular distance from the axis ; there will then be some relation
between cr and q, equivalent to the equation of the meridian curve! Then

ds2 = dcr2 + q2 d<j>2

where dO = — , so that 0 is a function of only one variable, the parameter of

the point regarded as a point on the meridian curve. Here n = <£ ; and so
the relation, which establishes the law of conformation between the plane
and the surface in the most general form, is

and the magnification ra is given by

Evidently the lines on the plane, which correspond to meridians of

* A discussion is given by Gauss, Ges. Werke, t. iv, pp. 211 — 216, of the corresponding result
when neither of the surfaces is plane.

255.] OF SURFACES OF REVOLUTION 497

longitude, are given by the elimination of 6, and the lines on the plane,
which correspond to parallels of latitude, are given by the elimination of <£,
between the equations

- id} +fl (</> — id)

Ex. 1. On the surface of revolution, let
where m, q, a- have the significations in the text ; shew that $ and ^ satisfy the equation

I \ i / \ A

where z^ z2 are the conjugate complexes x±_iy in the plane. (Korkine.)

2. Prove that, in a plane map of a surface of revolution, the curvature of a

"r\ / 1 \

meridian at a point 6 is g-r ( — J and the curvature of a parallel of latitude at a point $
. Hence shew that, if the meridians and the parallels of latitude become

circles on the plane map given by

!=/(*+«»),

the function / and the conjugate function /x must satisfy the relation

where {/, /*} is the Schwarzian derivative. (Lagrange.)

Ex. 3. A plane map is made of a surface of revolution so that the meridians and the
parallels of latitude are circles. Shew that, if (r, a) be the polar coordinates of a point on
the map determined by the point (<9, $) on the surface, then

-= - 2ac {ae2c8 cos 2 (c<£ +g) + bcos(g + k)},

sin a 2c9 •

= 2ac{ae sin2(c(f)+g) + bsm(g + k)},

V

where a, b, c, g, h are constants.

Prove also that the centres of all the meridians lie on one straight line and that the
centres of all the parallels of latitude lie on a perpendicular straight line. (Lagrange.)

256. The surfaces of revolution which occur most frequently in this
connection are the sphere and the prolate spheroid.

In the case of the sphere, the natural parameter of a point on a great-
circle meridian is the latitude X. We then have da = ad\, where a is the
radius ; and q = a cos X, so that

ds* = a2 d\* + a2 cos2X dp

— a? cos2X
where sech S = cos X. Hence we have

and the magnification m is given by

ma cos X = {/' (<£ +
F. 32

498 PROJECTIONS [256.

There are two forms of/ which are of special importance in representations
of spherical surfaces.

First, let /(/A) = kp, where k is a real constant ; then

and therefore X = k(j>, Y =k^ = k sech"1 (cos X) ;

that is, the meridians and the parallels of latitude are straight lines,
necessarily perpendicular to each other, because angles are preserved. The
meridians are equidistant from one another ; the distance between two
parallels of latitude, lying on the same side of the equator and having
a given difference of latitude, increases from the equator. We have
/' (</> + fe) = k =// (0 - ^) ; and therefore

k

m= — sec X,
a

or the map is uniformly magnified along a parallel of latitude with a
magnification which increases very rapidly towards the pole. This map is
known as Mercators Projection.

Secondly, let/(/u,) = keic^, where k and c are real constants ; then

X + iY= ke{cti>+is» = ke~c* (cos c(f> + i sin c</>),
and therefore X = ke~c* cos c<j> and Y = ke~c^ sin c</>.

For the magnification, we have

/' (</> + 1£) = ickeic^+l^ and // (<£ - iS)
so that met cos X = cke~c<^,

ck , ck (1 - sin

or m = —e~°* sec X =

— r- — : — ^ 1( — .
a a (1 +smX)J(c+1'

The most frequent case is that in which c = 1. Then the meridians are
represented by the concurrent straight lines

Y = X tan (f) ;
the parallels of latitude are represented by the concentric circles

,

1 + sin X

the common centre of the circles being the point of concurrence of the
lines ; and the magnification is

k

a (1 + sin X) '

This map is known as the stereographic projection : the South pole being the
pole of projection.

It is convenient to take the equatorial plane for the plane of z : the
direction which, in that plane, is usually positive for the measurement of

256.] OF SPHERES 499

longitude, is negative for ordinary measurement of trigonometrical angles. If
we project on the equatorial plane, we have

which gives a stereographic projection.

Ex. 1. Prove that, if x, y, z be the coordinates of any point on a sphere of radius a and
centre the origin, every plane representation of the sphere is included in the equation

for varying forms of the function /.

Ex. 2. Shew that rhumb-lines (loxodromes) on a sphere become straight lines in
Mercator's projection and equiangular spirals in a stereographic projection.

Ex. 3. A great circle cuts the meridian of reference (0 = 0) in latitude a at an angle a;
shew that the corresponding curve in the stereographic projection is the circle

( X + k tan a)2 + ( Y+ k cot a sec a)2 = k2 sec2 a cosec2 a.

Ex. 4. A small circle of angular radius r on the sphere has its centre in latitude c and
longitude a ; shew that the corresponding curve in the stereographic projection is the

circle

/ &cosccosa\2 /„ & cose sin a\2_ k2sin2r

\ cos r + sin c) \ cos r + sin c) (cos r + sin c)2 '

The less frequent case is that in which the constant c is allowed to remain

, in the function for the purpose of satisfying some useful condition. One

such condition is assigned by making the magnification the same at the

points of highest and of lowest latitude on the map. If these latitudes be

X1} X2, then

(1 — sin Xj)^0"1' _ (1 — sin X-j)^""1'
(1 + sin X,)*^1) ~ (1 + sin X2)i<c+1> '

/I— smXi\ , , ^/1-f-sinX^

o 1

so that c —

I - sin XA /I + sinX

This representation is used for star-maps : it has the advantage of leaving
the magnification almost symmetrical with respect to the centre of the map.

Ex. Prove that the magnification is a minimum at points in latitude arc sin c.

Shew that, if the map be that of a belt between latitudes 30° and 60°, the magnification
is a minimum in latitude 45° 40' 50"; and find the ratio of the greatest and the least
magnifications.

Note. Of the memoirs which treat of the construction of maps of surfaces
as a special question, the most important are those of Lagrange* and

* Nouv. Mem. de VAcad. Roy. de Berlin (1779). There are two memoirs : they occur in his
collected works, t. iv, pp. 635 — 692.

32—2

500 MAPS [256.

Gauss*. Lagrange, after stating the contributions of Lambert and of Euler,
obtains a solution, which can be ap plied to any surface of revolution ; and
he makes important applications to the sphere and the spheroid. Gauss
discusses the question in a more general manner and solves the question
for the conformal representation of any two surfaces upon each other, but
without giving a single reference to Lagrange's work : the solution is worked
out for some particular problems and it is applied, in subsequent memoirs f,
to geodesy. Other papers which may be consulted are those of Bonnet J,
Jacobi§, Korkinell, and Von der MuhllH; and there is also a treatise by
Hera**.

But after the appearance of Riemann's dissertation 'f-f', the question
ceased to have the special application originally assigned to it ; it has
gradually become a part of the theory of functions. The general development
will be discussed in the next chapter, the remainder of the present chapter
being devoted to some special instances of functional relations between w and
z and their geometrical representations.

The following three examples give the conformal representation of three surfaces upon
a plane.

Ex. 1. A point on an oblate spheroid is determined by its longitude I and its
geographical latitude p.. Shew that the surface will be conformally represented upon a
plane by the equation

for any form of the function / ; where sech <£ = cos p, and e is the eccentricity of the
meridian.

Also shew that, if the function / be taken in the form f(u)—te, the meridians in
the map are concurrent straight lines, and the parallels of latitude concentric circles ; and
that the magnification is stationary at points in geographical latitude arc sin c. (Gauss.)

Ex. 2. Let the semi-axes of an ellipsoid be denoted by p, (p2 — ft2) , (p2-c2) in
descending order of magnitude. Shew that the surface will be conformally represented
upon a plane by the equation

,,. . . , , ,

X+il=f U(w + iv) + £log — - - (-J-. - (\
J [ °e(u-a}Q(iv-a))

for any form of the function /; where u and v are expressed in terms of the elliptic
coordinates p15 p2 °f a point on the surface by the equations

* Schumacher's Astr. Abh. (1825) ; Ges. Werke, t. iv, pp. 189—216.
t Gott. Abh., t. ii, (1844), ib., t. iii, (1847) ; Ges. Werke, t. iv, pp. 259—340.
t Liouville, t. xvii, (1852), pp. 301—340.

§ Crelle, t. lix, (1861), pp. 74—88; Ges. Werke, t. ii, pp. 399—416.
|| Math. Ann., t. xxxv, (1890), pp. 588—604.
IT Crelle, t. Ixix, (1868), pp. 264—285.

** Lehrbuch der Landkartcnprojectionen, (Leipzig, Teubner, 1885).

ft "Grundlagen fur eine allgemeine Theorie der Functionen einer veranderlichen complexen
Grosse," Gottingen, 1851 ; Ges. Werke, pp. 3 — 45, especially § 21.

256.] EXAMPLES 501

p /c2 — 62\J

the modulus is ( -5 — ^ ) , the constant a is given by
c \p2-o2/ '

b = c dn a,
and the value of the constant h is tn a dn a — /? (a). (Jacobi.)

/ftp. 3. The circular section of an anchor-ring by a plane through the axis subtends an
angle rr — 2e at the centre of the ring, and the position of any point oh such a section is
determined by I, the longitude of the section, and by A, the angle between the radius from
the centre of the section to the point and the line from the centre of the section to the
centre of the ring.

Shew that, by means of the equations

tan |X =tan |e tan (uy tan e),

the surface of the anchor-ring is conformally represented on the area of a rectangle whose
sides are 1 and cot «. (Klein.)

257. It was pointed out that the conformation of surfaces is obtained by
a relation

and therefore that the conformation of planes is obtained by a relation

w=f(z),
whatever be the form of the function /, or by a relation

4> (w, z) = 0,

whatever be the form of the function <£. Some examples of this conformal
representation of planes will now be considered ; in each of them the
representation is such that one point of one area corresponds to one (and
only one) point of the other.

Ex. 1. Consider the correspondence of the two planes represented by
(a - 6) w2 - 22^ + (a + 6) = 0,

that is, 22 = (a — V) w-\ -- .

w

Let r and 6 be the coordinates of any point in the w-plane : and x, y the coordinates of
any point in the s-plane : then

2# = |~(a - 6) r + -— 1 cos 6, 2y = [~(a - b} r - ~+~ 1 sin 6.

Hence the 2-curves, corresponding to circles in the w-plane having the origin for their
common centre, are confocal ellipses, 2c being the distance between the foci, where
c2 = er,2 — fc2 : and the 2-curves, corresponding to straight lines in the ?0-plane passing
through the origin, are the confocal hyperbolas, a result to be expected, because the
orthogonal intersections must be maintained.

Evidently the interior of a w-circle, of radius unity and centre the origin, is, by the
above relation, transformed into the part of the s-plane which lies outside the ellipse
#2/«'2 + ?/2/i-==l, the w-circumfereiice being transformed into the s-ellip.se.

Ex. 2. Consider the correspondence implied by the relation

,_i /2A' \ , . , . , , 2/i

k J w=sn I — z )=sn /, where .r'-f-?,w'=s'= - z,
V«T / f

502 RECTANGLE AND CIRCLE [257.

with the usual notation of elliptic functions. Taking w = X+iY, we have

sn x' en iy1 dn iy' + sn iy' en x1 dn x1
1 — k2 sn2 x' sn2 iy'

Let/=±pT': then sniy'=±-j-, flo^-V, dniy = <sfi+~*, «° that

„ ( 1 + k} sn x1 „ en x1 dn of

whence A = \ , „ , , I = ± ^—j — =- , ,

1 + k sn J ar ' 1 + « sn2 .r

and therefore A'2 + J ' 2 = 1 ,

which is the curve in the w-plane corresponding to the lines y'= ±.^K' in the .s'-plane,

77 ' 1\!

that is, to the lines 11= + —,. in the 2-plane.
— 4yt

When y= -f and ^/ lies between A" and - K, that is, .r lies between %w and - |TT, then
Y is positive and X varies from 1 to - 1 ; so that the actual curve corresponding to the
line y = -~rj^ is the half of the circumference on the positive side of the axis of X. Simi
larly the actual curve corresponding to the line y= —~-rj7 is the half of the circumference
on the negative side of that axis.

The curve hereby suggested for the 2-plane is a rectangle, with sides x=+. |TT,

IT K'

y= + - -jr . To obtain the w-curve corresponding to X = \TC, that is, to x' = K, we have

cniy'
dn iy' '

so that F=0 and X=i

Now y' varies from \K' through 0 to - \K' : hence X varies from 1 to $ and back
from $ to 1. Similarly, the curve corresponding to x= -£TT,
that is, to x' = — K, is part of the axis of X repeated from
-1 to —$ and back from -$ to — 1.

Hence the area in the w-plane, corresponding to the rect
angle in the 2-plane, is a circle of radius unity with two diametral
slits from the circumference cut inwards, each to a distance $
from the centre.

The boundary of this simply connected area is the homo-
logue of the boundary of the 2-rectangle given by x= ±577,

y = ± -jje : the analysis shews that the two interiors corre
spond*. And the sudden change in the direction of motion of the w-point at the inner
extremity of each slit, while z moves continuously along a side of the rectangle, is due to
the fact that dwjdz vanishes there, so that the inference of § 9 cannot be made at this
point. (See also Ex. 10.)

* For details of corresponding curves in the interiors of the two areas, see Siebeck, Crelle, t.
Ivii, (1860), pp. 359—370; ib., t. lix, (1861), pp. 173—184 : Holzmiiller, treatise cited (p. 2, note),
pp. 256—263 : Cayley, Camb. Phil. Trans., vol. xiv, (1889), pp. 484—494.

257.]

RECTANGLE AND ELLIPSE

503

Corollary. We pass at once from the rectangle to a square, by assuming K ' = 2K ; then
&=(»/2- I)2, and the corresponding modifications are easily made.

Ex. 3. Shew that, if 2 = sn2(Jw, k) where w = u+iv, then the curves u = constant,
v •= constant, are confocal Cartesian ovals whose equations may be written in the form

i\ - r dn (u, k) = en (u, £), rt+r dn (vi, k') = en (vi, &')>
where r and r± denote the distances from the foci 2 = 0 and z=\.

If r2 denote the distance of a point from the third focus z = , , find the corresponding

K

equations connecting r, r2; and r1}.r2.

Shew that the curves u = K, v = K' are circles, and that the outer and the inner branches
of an oval are given by u and 2K-u, or by v and ZfC — v. (Math. Trip. Part n, 1891.)

Ex. 4. The ?0-plane is conformally represented on the 2-plane by the equation

=
c~ \\-wJ '

where h and c arc real positive constants.

Shew that, if an area be chosen in the w-plane included within a circle, centre the
origin and radius unity, and otherwise bounded by two circles centres 1 and - 1 (so that
its whole boundary consists of four circular arcs), then the corresponding area in the
2-planc is a portion of a ring, bounded by two circles, of radii ceh and ce~h and centre the
origin, and by two lines each passing from one circle to the other.

Prove that, when the semi-circles in the w-plane are very small, so as merely to
exclude the points 1 and — 1 from the circular area and boundary, the corresponding
2-figure is the ring with a single slit along the axis of real quantities *.

Ex. 5. Consider the correspondence implied by the relation

Taking w = X+iY, we have

x + iy = c sin (X+i F)

= c sin JTcosh F+ ic cos Jf sinh F,

so that a? = c sin X cosh F, y = c cos X sinh F.

When F is constant, then z describes the curves

x2- y2 _

c2 cosh2 F c2 sinhnr" '

which, for different values of F, are confocal ellipses.

Now take a rectangle lying between X= ±|TT, Y=±\. For all values of X,
cos JT is positive : hence when F= +\ y
is positive and x varies from ccoshX to
-c cosh X, that is, the half of the ellipse on
the positive side of the axis of y is covered.

Let X= — £?r : then

y = 0 and x= -ccosh F.
As F varies from + X through 0 to - X
along the side of the rectangle, x passes
from B to H (the focus) and back from H
to B.

* See reference, p. 431, note.

504 ELLIPSE AND CIRCLE [257.

When Y= - X, then z describes the half of the ellipse on the negative side of the axis of
y: when X=+\TT, then y=0, # = ccoshF, so that z passes from A to S (a focus) and
back from S to A.

Hence the 2-curve corresponding to the contour of the w-rectangle is the ellipse
with two slits from the extremities of the major axis each to the nearer focus : the
analytical relations shew that the two interiors correspond.

Ex. 6. Consider the correspondence implied by the relation

,_. /2/T . .z\ (%K \

k fw = sn — sm~1-)=sn — f],

\* Cj \7T V

From Ex. 2, it follows that the interior of a w-circle, centre the origin and radius
unity, corresponds to the interior of the f-rectangle bounded by x= ±^ir, y=±—^,
provided two diametral slits be made in the w-circle along the axis of x to distances
1— £* from the circumference ; and, from Ex. 5, it follows that the same f -rectangle is
transformed into the interior of the 2-ellipse

where a = c cosher and 6 = csinh- „. , provided two slits be made in the elliptical area
along the major axis from the curve each to the nearer focus.

Thus, by means of the rectangle, the interiors of the slit w-circle and the slit 2-ellipse
are shewn to be conformal areas.

But the lines of the two slits are conformally equivalent by the above equation. For
the elliptical slit on the positive side of the axis of x extends from x = c to # = ccoshX,

where ^ — -T^> and it has been described in both directions : we thus have

2 = c cosh £,
where /3 passes from 0 to X and back from X to 0. Hence

* Z

sin ~ 1 - = sin ~ 1 (cosh /3) =
so that the corresponding w-curve is given by

Then, when /3 assumes its values, w passes from 1 to $ and back from £* to 1, that is,
w describes the circular slit on the positive side of the axis of X.

Similarly for the two slits on the negative side of the axis of real quantities. Thus
the two slits may be obliterated : and the whole interior of the w-circle can be represented
on the interior of the 2-ellipse.

From the equations denning a and b, it follows that

in the Jacobian notation; and c2 = «2-

257.] PARABOLA AND CIRCLE 505

Combining the results of Ex. 1 and Ex. 6 we have the theorem* : —

T lie part of the z-plane, which lies outside the ellipse x2/a2+y2/b'* = l, is transformed
into the interior of a w-circle, of radius unity and centre the origin, by the relation

and the part of the z-plane, which lies inside the same ellipse, is transformed into the interior
of the same w-circle by the relation

where the Jacobian constant q which determines the constants of the elliptic functions, is
given by

Ex. 1. Consider the correspondence implied by the relation

(w + I)*z = 4.

When w describes a circle, of radius unity and centre the origin, then w = e^>i: so that,
if r and 6 be the coordinates of z, we have

- (cos 6-i sin 0) = (1 +«*')*,

/ A fi\

(cos-- i sin- ) = 1 + e** = 1 + cos <£ + a sin <£.
r \ " * /

„ /2 6 A2 ,4 . 90

Hence ( -— cos - - 1 I + - sin2 - = 1,

Wr 2 / r 2

a
that is, r cos2 ~ = 1,

shewing that z then describes a parabola, having its focus at the origin and its latus
rectum equal to 4.

Take curves outside the parabola given by

r= 2sec2-
where p. is a constant ^ 1. Then

-p. = - cos ^0,
\'r A1

so that w -f 1 = -— e ~ ^ = - e~^° cos $0 ;

\ r M

2 1

therefore A" + 1 = - cos2 i<9 = ( 1 + cos <9),

M M

Y= - - sin 6,

/ 1 \ 2 1

HO that LY+ 1--) +T2= i,

a series of circles touching at the point X= - 1, F=0, and (for p. varying from 1 to oo )
covering the whole of the interior of the w-circle, centre the origin and radius unity.

* Schwarz, Oes. Werke, t. ii, pp. 77, 78, 102—107, 141.

506 CIRCLE AND HALF-PLANE [257.

Hence, by means of the relation (w + l)22=4, the exterior of the 2-space bounded by
the parabola is transformed into the interior of the w-space bounded by the circle.

Ex. 8. Consider the correspondence implied by the relation

w = tan2 ( JITS*).
We have

so that, if w + l=Re&l, <w = |TJT* cos |0, v = fynr^s,m%6, then

2R~l cos 8 — 1 = cos u cosh v,
2/i!~1sin0 = sinwsinhv.

The w-curves, corresponding to the confocal parabolas in the z-plane, are
(2 cos e -J

_ =

sin2 u

If iK^rr, then 2R~l cos G> 1, that is, R<2 cos 9 ; while, if U>%TT, we have R>2 cos 0.

It thus appears that the 2-space lying within the parabola U = %TT, that is, r cos2 £0=1,
is transformed into the interior of a w-circle, centre the origin and radius unity, by means
of the relation

By the two relations* in Ex. 7 and Ex. 8, the spaces within and without the parabola
are conformally represented on the interior of a circle.

Ex. 9. Consider the relation

_ _
^•\-^l} '

then, if z — x+iy and w = X+iYy we have

. !-

x+iy-

When w describes the whole of the axis of X from — QO to -4- oo , so that we can take

X=ta,nd). Y=0, where 0 varies from -^ to +5', we have ,r = cos2<£, 3/ = sin2^ ; and z

2t 2t

describes the whole circumference of a circle, centre the origin and radius 1. For internal
points of this circle \-xi-yL is positive : it is equal to 4F-r- {Z2 + (1 + F)2}, and there
fore the positive half of the w-plane is the area conformal with the interior of the circle,
of radius unity and centre the origin, in the s-plane.

Ex. 10. Again, consider a relation

. ^ (x2 + V2 - c2)2 - 4c%2 + 4ic0 (c2 - a? - f)
We have

„_ ±cx (c2 — xz —

=

Let x=0, so that Y—0 ; then

* Schwarz, Ges. Werke, t. ii, p. 146.

257.]

SEMI-CIRCLE AND HALF-PLANE

507

As z passes from A to B (where OA = OB = c), then y changes from -c to +c, and X
changes continuously from +00 to 0.

Let #2+y2-c2=0, so that F=0 ; then

A 0--2 /•» — fit

-rr TTtV ^ </ • O 1 /I

where y = c cos $. Hence, as z describes the semi-circular
arc Z?(L4, the angle 0 varies from 0 to n and X changes
from 0 to — QO .

(The whole axis of X is the equivalent of AOBCA ; and
at the w-origin, corresponding to J5, there is no sudden
change of direction through £TT. The result is apparently
in contradiction to § 9 : the explanation is due to the
dw

Fig. 88.

fact that -7— cO at B, and the inference of § 9 cannot be made. Similarly for A where

d/z

-=- is infinite. See also Ex. 2.)
dz

For any point lying within the z-semi -circle, both x and e2-A<2-#2 are positive, so
that Y is positive. Hence by the relation

(z — ic
=

the interior of the z-semi-circle is conformally represented on the positive half of the
w-plane.

It is easy to infer that the positive half of the to-plane is the conformal equivalent

of

(z — ic\2
(i) the interior of the semi-circle ACS A by the relation w = ( —- :

\

(") .......................................... CBDC

(iii) ......... BDAB

(iv) .......................................... DACD

And, by combination with the result of Ex. 9, it follows that the relation

. fz — ic\2

"

I +

s2-c2-2cz

conformally represents the interior of the z-semi-circle ACBA on the interior of the
?0-circle, radius unity and centre the origin.

Similarly for the other cases.

Ex. 11. Find a figure in the z-plane, the area of which is conformally represented on

the positive half of the w-plane by

(z — ic\n
(i) w=zn, (ii) w={— T^-\ .

Ex. 12. Consider the relation

then

(ii) w=(— —^
\z+ic

w = ae** :
X= ae~y cos x, Y= ae~v sin x.

508

EXAMPLES OF CONFORMAL

[257.

The curves corresponding to y = constant are concentric circumferences; those corre
sponding to x = constant are concurrent straight lines.

As x ranges from 0 to far, both X and F are positive ; for a given value of x between
these limits, each of them ranges from 0 to oo , as y ranges from oo to - oo . As x ranges
from ^TT to TT, X is negative and Y is positive ; for a given value of x between these
limits, — X and Y range from 0 to oo , as y ranges from oo to — co .

Hence the portion of the 2-plane lying between y= -oo, y=ao, x = 0, X = TT, that is, a
rectangular strip of finite breadth and infinite length, is conformally represented by the

relation

w=aeiz

on the positive half of the w-plane. Combining this result with that in Ex. 9, we see that
the same strip is conformally represented on the area of a w-circle, centre the origin and

radius a, by means of the relation

w- 1

w + l

Note. It may be convenient to restate the various instances of areas in the z-plane,
bounded by simple curves, which can be conformally represented on the area of a
circle in the w-plane :

(i) The positive half of the z-plane ; Ex. 9.
(ii) An infinite strip of finite breadth; Ex. 9, Ex. 12.
(iii) Area without an ellipse ; Ex. 1.
(iv) Area within an ellipse ; Ex. 6.
(v) Area without a parabola ; Ex. 7.
(vi) Area within a parabola ; Ex. 8.
(vii) Area within a rectangle ; Ex. 2.

(viii) As will be seen, in § 258, any circle changes into itself by a proper homo-
graphic relation.

Ex. 13. Consider the correspondence implied by the relation

Then we have two values of w3, say w^3, w2s, where

1-2* 1+3*

1-**

Let z describe the axis of x, so that z=x.

When 0<.r<l, then w^ is real and less than
unity and w23 is real and greater than unity. Hence
drawing a circle in the ?«-plane, centre the origin
and radius 1, and six lines as diameters making angles
of ^TT with one another, and denoting a cube root of
1 by a, then, as z passes from 0 to 1 along the axis of x,

wl passes from A to 0,

w2 A to A' (at infinity),

awl C to 0,

a«>2 C to C' (at infinity),

aX E to 0,

a-w.2 E to E' (at infinity).

Fig. 89.

257.] REPRESENTATION 509

When l<A'<oo, then w^ is a real quantity changing continuously from 0 to -1, and
w23 is a real quantity changing continuously from - oo to -1. As z passes from 1 to oo
along the positive part of axis of X,

wv passes from 0 to F,

w2 B' (at infinity) to B,

aiol 0 to B,

aw2 D (at infinity) to D,

aX 0 to Z),

a2w2 F' (at infinity) to F.

Hence, as z describes the whole of the positive part of the axis of x, the branches of w
describe the whole of the three lines' -4'ZX, B'E', OF'.

When x is negative, we can take x= - tan2$, so that <£ varies from 0 to |TT. Then

1 + i tan (j>

so that, as z passes from 0 to - oo , u\ describes the arc of the circle from A to F, aivl the
arc from C to Z?, and a2w1 the arc from E to D. And then

so that u\2 describes the arc of the circle from A to B, aw2 the arc from C to D, and a2w.
the arc from E to F. Hence, as z describes the whole of the negative part of the axis of x,
the branches of w describe the whole of the circumference.

As z describes a line parallel to the axis of x and very near it on the positive side, the
paths traced by the branches are the dotted lines in the figures ; the six divisions in
which the symbols are placed, are the conformal representations by the six branches
of w of the positive half of the 2-plane*.

Ex. 14. When the variables are connected f by a relation

w= —

where $0 is the function which in coefficients is conjugate to 0, then the 2-circumference,
centre the origin and radius c, is transformed into the w-circumference, centre the
origin and radius c.

Taking w0 and z0 as the conjugate variables, we have

so that M>WO = -~

Now if z describe the circumference of a circle, centre the origin and radius c, we have

z=ce6t, z0 = ce~ei, zz0 = c2,
so that wwn = c2

shewing that w describes the circumference of a circle, centre the origin and radius c.

* Cayley, Camb. Phil. Trans., vol. xiii, (1880), pp. 30, 31.
t Cayley, Crelle, t. cvii, (1891), pp. 262—277.

510 EXAMPLES OF CONFORMAL [257.

To determine whether the internal area of the ^-circumference corresponds to the in
ternal area of the w-circumference, we take zz0 = c2 — e, where e is small . Then

, , ,. „ ( , , nif\ ( f (f)' (z)} ( ( <f>0' (20)1
therefore wiv0=c* I l+-s- I •?! -X7-(M1 — ~ jr^f

\ c2/ 1 ^o0(2)J 1 ^ 00(^0)J

so that the interior of the z-circumference finds its conformal correspondent in the
interior or in the exterior of the w-circurnference according as

,

taken along the circumference.

The simplest case is that in which (£ (z) is of degree m, so that it can be resolved
into m factors, say $ (z) = A (z— a}(z-fi}...(z-6) : then

and

But the converse of the result obtained — that to the w-circumference there corresponds
the 2-circumference — is not complete unless the correspondence is (1, 1). Other curves
which are real — they may be, but are not necessarily, circles — and imaginary curves enter
into the complete analytical representation on the g-plane corresponding to the w-circum-
ference, of centre the origin and radius c on the w-plane.

Ex. 15. Discuss the 2-curves corresponding to \w =1, determined by

zfz—Jty /n , .

w = ->- — =^ . (Cayley. )

i-vsi

Ex. 16. Consider the relation

4 22-2+l3

'=27
We have w-w0 = —

2tt

The function on the right-hand side, being connected with the expressions for the six
anharmonic ratios of four points in terms of any one ratio, vanishes for

z=zoi —•> l-^oj i~_~^> "0~ITi ' "^ '

so that w-iv0 = —- 5 o

27 (r—S)

Hence, taking w = X+iY, z =

r=^ 4 2i<
1 ~27

257.]

REPRESENTATION

511

Hence it appears that, when F=0, so that w traces the axis of real quantities in its
own plane, the z-variable traces the curves

that is, two straight lines and two circles in its
own plane.

In order to determine the parts of the 2-plane
that correspond to the positive part of the w-plane,
it is sufficient to take Y equal to a small positive
quantity and determine the corresponding sign of
y. Let

where Y (and therefore y} is small : then, to a first
approximation,

97 r^fr— ~\}3

41 Jj It*/ — A I

Fig. 90.

_ _

~ T (2a? - 1) (x + 1 ) (x - 2) (x* - x + 1 )* '

ind the sign of ^ determines whether the part on the positive or negative side of the axis
of x is to be taken.

When x< - 1, /j. is negative; z lies below the axis of x. When x is in AO, so that
x> - 1<0, fj, is positive ; z lies above. When x is in OB, so that x>0<^, /u is negative ;
z lies below. When x is in BC, so that #>£<!, /u. is positive ; z lies above. When x is
in CD, so that ^>1<2, p, is negative ; s lies below. And, lastly, when x is beyond Z), so
that x>2, p, is positive and z lies above the axis of real quantities. The parts are indicated
by the shading in fig. 90.

It is easy to see that w=Q, for z = P, Q; that w=\, for z — A, B, D ; and that w=oo,
for z=0, C. The zero value of w is of triple occurrence for each of the points P and Q;
the unit-value and the infinite value are of double occurrence for their respective points*.

Note. It is easy to see that figures 89 and 90 are two different stereographic projections
of the same configuration of lines on a sphere (§ 277, I, ft = 3), so that the relations in
Ex. 13 and Ex. 16 may be regarded as equivalent.

Ex. 17. Find, in the same way, the curves in the js-plane, which are the conformal
representation of the axis of X in the w-plane by the relation t

Ex. 18. Shew that, by the relation

the lines, x — constant in the 2-plane, are transformed into a series of confocal lemnis-
cates in the w-plane ; and that, by the relation

z2 (c2 + w2 — 1 ) = cw2,

where c is a real positive constant greater than unity, the interior of a 2-circle, centre
the origin and radius unity, is transformed into the interior of the lemniscate RR'=c2

in the w-plane, where It and It' are the distances of a point from the foci (1, 0) and

(-1,0). (Weber.)

See Klein-Fricke, vol. i, p. 70.

t See Klein-Fricke, vol. i, p. 75.

512 HOMOGRAPHIC [258.

258. The preceding examples* may be sufficient to indicate the kind of
correlation between two planes or assigned portions of two planes that is
provided in the conformal representation determined by a relation </> (w, z) = 0
connecting the complex variables of the planes. We shall consider only one
more instance ; it is at once the simplest and functionally the most important
of all "f*. The equation, jvvhich characterises it, is linear in both variables ; and
so it can be brought into the form

_ az + b
cz + d'

where a, b, c, d are constants : it is called a homographic transformation, some
times a homographic or a linear substitution.

Taking first the more limited form

w=1' . ' i

and writing w = Rei&, z —rei6, p = k^{, we have

Er = !<?, ® + 9 = 27, that is, ©-7 = 7-0,

and therefore the new w-locus will be obtained from the old ^-locus by
turning the plane through two right angles round the line 7 through the
origin, and inverting the displaced locus relative to the origin. The first
of these processes is a reflexion in the line 7 ; and therefore the geometrical
change represented by WZ — JM is a combination of reflexion and inversion.

A straight line not through the origin and a circle through the origin are
corresponding inverses ; a circle not through the origin inverts into another
circle not through the origin and it may invert into itself; and so on.

Taking now the general form, we have

a ad — be

w — = • -T- ,

c -2 a\
\ c)

or transforming the origins to the points - and in the w- and the ^-planes

c c

respectively, and denoting — • — by p, we have WZ = p, that is, the former

C"

case. Hence, to find the w-locus which is obtained through the transforma
tion of a 2-locus by the general relation, we must transfer the origin to - - ,
turn the plane through two right angles round a line through the new origin

* Many others will be found in Holzmiiller's treatise, already cited, which contains ample
references to the literature of the subject.

t For the succeeding properties, see Klein, Math. Ann., t. xiv, pp. 120 — 124, ib., t. xxi,
pp. 170—173 ; Poincar<§, Acta Math., t. i, pp. 1—6 ; Klein-Fricke, Elliptische Modulfunctionen,
vol. i, pp. 163 et seq. They are developed geometrically by Mobius, Ges. Werke, t. ii, pp. 189—204,
205—217, 243—314.

258.] TRANSFORMATION 513

whose angular coordinate is £ arg. ( -• — - — 1 , invert the locus in the displaced

\ c /

position with a constant of inversion equal to

be — ad

, and then displace the

origin to the point — . Hence a circle will be changed into a circle by a
c

» homographic transformation unless it be changed into a straight line ; and
a straight line will be changed into a circle by a homographic transformation
unless it be changed into a straight line.

The result can also be obtained analytically as follows ; the formulas
relating to the circle will be useful subsequently.

A circle, whose centre is the point (a, /9) and whose radius is r, can be
expressed in the form

(z — a. — /3i) (z0 — a + @i) = r2,

or zz0 + 0z + 0^ + 7 = 0,

where —0 = a-fti, — #„ = a + fti, 7 = 00{} — r2. Conversely, this equation
represents a circle, when 0 and 00 are conjugate imaginaries and 7 is real ; its
centre is at the point — £ (0 + 00), ^i (0 — 00), and its radius is (000 — 7)*.

When the circle is subjected to the homographic transformation

_ az + b
~ cz + d'

— dw + b 1.1- — d0w0 + b0

we have z = and therefore z0 = - — .

cw — a cuw0 — a0

Substituting these values, the relation between w and w0 is

S'ww0 + 0'w + 00'w0 + 7' = 0,
where S' = dd0 — 0dc0 — 00cd0 + ycc0 ,

& = — bud + 0aud + 0ucb0 —

00 — — bd0 + 0cJ} + 0^ad0 — >

y = bb0 - 0a()b - 00ab0 + <yaa0 :

here 8' and 7' are real, and 0' and 00' are conjugate imaginaries; therefore the
equation between w and wu represents a circle.

Ex. A circle, of radius r and centre at the point (e, /), in the z-plane is transformed
into a circle in the w-plane, by the homographic substitution

az + b
~cz+d '
shew that the radivis of the new circle is

r_ ad- be
A c2

where A = (<r cos ft + e)2 + (a- sin /3 +/)2 - r\

F 33

514 CANONICAL FORM [258.

and o-, ft are the modulus and the argument respectively of - . Find the coordinates of
the centre of the ^-circle.

Moreover, since there are three independent constants in the general
homographic transformation, they may be chosen so as to transform any three
assigned ^-points into any three assigned w-points. And three points on a
circle uniquely determine a circle : hence any circle' can be transformed into
any other circle (or into itself} by a properly chosen homographic transforma
tion. The choice of transformation can be made in an infinite number of ways :
for three points on the circle can be chosen in an infinite number of ways.

A relation which changes the three points z1} z.2> zs into the three points
wly w2, w3 is evidently

(w - w,) (ws - w3) _ (z - zj Q2 - Q
(w - w2) (w, - ws) ~ (z - z2) (zj. - za) '

Hence this equation, or any one of the other five forms of changing the three
points z1} zz, zz into the three points wltw3, w3 in any order of correspondence,
is a homographic transformation changing the circle through zl} z?, z3 into the
circle through wlt w2, w3.

It has been seen that a transformation of the form w=f(z) does not
affect angles : so that two circles cutting at any angle are transformed by

w = into two others cutting at the same angle. Hence* a plane crescent,

cz + d

of any angle, can be transformed into any other crescent, of the same angle.

The expression of homographic transformations can be modified, so as to
exhibit a form which is important for such transformations as are periodic.

If we assume that w and z are two points in the same plane, then there
will in general be two different points which are unaltered by the transfor
mation ; they are called the fixed (or double) points of the transformation.
These fixed points are evidently given by the quadratic equation

au + b

u =

cu + d'

that is, ci<? — (a — d)u-b = Q.

Let the points be a and /3, and let M denote (d — a)2 + 4&c ; then

2ca = a - d + M±,
If, then, the points be distinct, we have

w — a. _ (z — a)(a — ca) _ ^ z — a
~~

z-/3'

Kirchhoff, Vorlesungen iiber matliematisclie PJiysik, i, p. 286.

258.] OF HOMOGRAPHIC TRANSFORMATION 515

a — ca.
where K =

,
a-cp a + d + M *

1 y _ (a+d)2

and therefore I v /i " /^,

The quantity K is called the multiplier of the substitution.

If there be a ^-curve in the plane passing through a, the w-curve which
arises from it through the linear substitution also passes through a. To find
the angle at which the ^-curve and the w-curve intersect, we have w = a + $w,

z = a + Sz : and then

Sw = JKSz,

so that the inclination of the tangents at the point is the argument of K.
Similarly, if a ^-curve pass through /3, the angle between the tangents to the
w curve and the ^-curve is supplementary to the argument of K.

The form of the substitution now obtained evidently admits of reapplica-
tion ; if zn be the variable after the substitution has been applied n times, (so
that z0 = z, zl = w), we have

zn-/3 z-p-

The condition that the transformation should be periodic of the nth order
is that zn — z and therefore that Kn = 1 ; hence

Sir
(a + d)2 = 4 (ad - be) cos2 — ,

where s is any integer different from zero and prime to n; K cannot be
purely real, and, in general, M is not a real positive quantity. The
various substitutions that arise through different values of s are so related
that, if points zly z.2,..., zn be given by the continued application of one
substitution through its period, the same points are given in a different
cyclical order by the continued application of the other substitution through
its period.

Ex. 1. The value of zn has been given by Cayley in the form

obtain this expression.

Ex. 2. Periodic substitutions can be applied, in connection with Kirchhoffs result
that a plane crescent can be transformed into another plane crescent of the same angle ;
the plane can be divided into a limited number of regions when the angle of the crescent
is commensurable with TT.

Let ACBDA be a circle of radius unity, having its centre at the origin: draw the
diameter AB along the axis of y. Then the semi-circle ACB can be regarded as a plane

33—2

516

EXAMPLES

[258.

crescent, of angle |TT ; and the semi-circle ABD as another, of the same angle. Hence
they can be transformed into one another.

We can effect the transformation most simply by taking A (=i) and B(= — i] as the
fixed points of the substitution, which then has the form

w + i z + i'

The line AB for the w-curve is transformed from the z-circular arc ACB: these curves
cut at an angle ^TT, which is therefore the argument of K. Considerations of symmetry
shew that the 2-point C on the axis of x can be transformed into the w-origiu, so that

whence K=i, so that the substitution is

w — i .z

. = ^

It is periodic of order 4, as might be expected : when simplified, it takes the form

1+2

w = - - .

1-2

The above figure shews the effect of repeated application of the substitution through
a period. The first application changes the interior of ACB A into the interior of ABD A :
by a second application, the latter area is transformed into the area on the positive side of
the axis of y lying without the semi-circle ADB ; by a third application, the latter area is
transformed into the area on the negative side of the axis of y lying without the semi
circle ACB; and by a fourth application, completing the period, the latter area is
transformed into the interior of ACB A, the initial area.

The other lines in the figure correspond in the respective areas.

258.] HOMOGRAPHIC SUBSTITUTIONS 517

Ex. 3. Show that, if the plane crescent of the preceding example have an angle
of -IT instead of |TT but still have +i and —i for its angular points, then the substitution

z + t
w—-

where t denotes tan — , is a periodic substitution of order 2re which, by repeated appli
cation through a period to the area of the crescent, divides the plane into 2n regions, all
but two of which must be crescent in form. Under what circumstances will all the 2n
regions be crescent in form ?

Note. The formula in the text may be regarded as giving the nth power of a substi
tution. The form of the substitution obtained is equally effective for giving the nth root
of a substitution : all that is necessary is to express K in the form pee\ and the nth
root is then

Z- a

GiZ-a.

~n

259. Homographic substitutions are divided into various classes, according
to the fixed points and the value of the multiplier. As the quantities a, b,
c, d can be modified, by the association of an arbitrary factor with each of
them without altering the substitution, we may assume that ad — be = 1 ;
we shall suppose that all substitutions are taken in such a form that their
coefficients satisfy this relation. Figures which, by them, are transformed
into one another are called congruent figures.

If the fixed points of the substitution coincide, it is called* a parabolic
substitution.

There are three classes of substitutions, which have distinct fixed points.
If the multiplier be a real positive quantity, the substitution is called
hyperbolic.

If the multiplier have its modulus equal to unity and its argument
different from zero, it is called elliptic.

If the multiplier have its modulus different from unity and its argument
different from zero, it is called loxodromic.

These definitions apply to all substitutions, whether their coefficients be
real or be complex constants ; when we consider only those substitutions,
which have real coefficients, only the first three classes occur. Such sub
stitutions are often called real.

The quadratic equation, which determines the common points of a real
substitution, has its coefficients real ; according as the roots of the quadratic
are imaginary, equal, or real, the real substitution will be proved to be
elliptic, parabolic, or hyperbolic respectively. For all of these, we take

c c

All these names are due to Klein : 1. c., p. 512, note.

518 PARABOLIC SUBSTITUTIONS [259.

which imply a transference of the respective origins along the respective axes
of real quantity ; and then

c2 x 4- iy

x — iy
~

F_ 1

so that — o /"o" v\ •

y c2 (x- + 2/2)

The axes of x and of X have been unaltered by any of the changes made in
the substitution ; and F, y have the same sign and vanish together ; hence
the effect of a real transformation is to conserve the axis of real quantities, by
transforming the half of the ^-plane above the axis of x into the half of the
w-plane above the axis of X.

A real transformation, which changes z into w, also changes z0 into tv0
(these being conjugate complexes). A circle, having its centre on the
axis of x and passing through a, ft, passes through a0, ft0 also: hence a
transformation, which changes a circle through a, ft with its centre on
the axis of x into one through 7, 8 with its centre on the axis of X, is

z — a. ft — a0 _ w — 7 8 — 70

z — a.0'ft — a. w—jo'B — y'

Ex. 1. Shew that, if this circle, through a, /3, a0, /30, cut the axis of x in h and k,
where h lies in @pQ and k in aa0, and if [a/3] denote ^-r . •s— , , a real quantity greater than

(1 — K J3 /i

1, then

* [a/3] (Poincare.)

2

(a-/30)O-«o)

J&;. 2. Prove that the magnification at any point, by a real substitution, is Yjy.

(Poincare.)

Ex. 3. Any z-circle, having its centre on the axis of x, is transformed by a real
substitution into a w-circle, having its centre on the axis of X.

Let the classes of real substitutions be considered in order.

(i) For real parabolic substitutions, the quadratic has equal roots : let
their common value be a, necessarily a real quantity, so that the fixed points
of the substitution coalesce into one on the axis of x. The quantity M is
then zero, so that (d + a)'2 = 4. We may, without loss of generality, take
d+a = 2. If both origins be removed to the point a, then, in the new
form, zero is a repeated root of the quadratic, so that 6 = 0, and a - d = 0.
Hence a = d = I, and the real substitution is

that is,*

or+11

If the origins be not removed to the point a, the form is ^-— ^ = ^—^

259.] ELLIPTIC SUBSTITUTIONS 519

The equations of transformation of real coordinates are

Ex. 1. A 2-circle passing through the origin is transformed, by a real parabolic substi
tution having the origin for its common point, into a w-circle, passing through the origin
and touching the z-circle : and a 2-circle, touching the axis of x at the origin, is trans
formed into itself.

Ex. 2. Let A be a circle touching the axis of x at the origin : and let c0 be the
extremity of its diameter through the origin. Let a real parabolic substitution, having
the origin for its common point, transform c0 into cls c± into c2, c2 into c3, and so on : all
these points being on the circumference of A.

Prove that the radii of the successive circles, which have their centres on the axis of x
and pass through the origin and clt the origin and c2, ... respectively, are in harmonic
progression, and that, if these circles be denoted by (715 (72, ..., then Ck is the locus of all
points ck arising through different initial circumferences A.

Ex. 3. What is the effect of the inverse substitution, applied as in Ex. 2 ?

Ex. 4. Shew that, if a curve of finite length be drawn so as to be nowhere infini-
tesimally near the axis of x, it can cut only a finite number of the circles C in Ex. 2.

(Note. All these results are due to Poincare.)

(ii) For real elliptic substitutions, a and ft are conjugate complexes ;
hence M is negative, so that

(d - a)2 + 46c < 0,

or (d + a)2 < 4 (ad - be) < 4.

The value of K, by using the relation ad — be = 1, is

It is easy to see that \K\ = 1 and that its argument is cos"1 {| (a + dj2 — 1}, so
that, if this angle be denoted by a-, we have

JT-«*

shewing that the substitution is elliptic.

It is evident that, if z describe a circle through a and ft, its centre being
therefore on the axis of x, then w also describes a circle through a and ft
cutting the ^-circle at an angle <r. The two curves together make a plane
crescent of angle a having o, ft for its angular points.

Ex. Shew that a real elliptic substitution transforms into itself any circumference,
which has its centre on a/3 produced and cuts the line a/3 harmonically. (Poincare.)

(iii) For real hyperbolic substitutions, the roots of the quadratic are real
and different ; hence the fixed points of the substitution are two (different)
points on the axis of x. The quantity M is positive, so that

(a + df > 4 :

520 HYPERBOLIC SUBSTITUTIONS [259.

we may evidently take a -f d > 2. Moreover K is real and positive, shewing
that the substitution is hyperbolic.

Taking one of the fixed points for origin and denoting by /the distance
of the other, we have 0 and /as the roots of

_ au + b

Us -. .

cu + d

with the conditions ad — be — 1, a + d > 2. Hence 6 = 0, a — d = cf, ad — 1,
K = -j ; then K is greater or is less than 1 according as cf is positive or is

negative. We shall take K > 1 as the normal case ; and then the sub
stitution is

az

cz + d'
with a > 1 > d, a + d > 2, ad — 1 .

Ex. 1. A 2-curve is drawn through either of the fixed points of a real hyperbolic
substitution : shew that the w-curve, into which it is changed by the substitution, touches
the 2-curve. Hence shew that any 2-circle through the two fixed points of the substi
tution is transformed into itself.

Ex. 2. Let A be a circle through the origin and the point /; and let c0 be the other
extremity of its diameter through /. Let a real hyperbolic substitution, having the origin
and / for its fixed points, transform c0 into c15 cx into c2, c2 into c3, and so on : all these
points being on the circumference of A.

Shew that the radius of a circle Cn, having its centre on the axis of x and passing
through cn and the origin, is

so that Cn is the locus of all the points cn arising through different initial circumferences
A . What is the limit towards which Cn tends as n becomes infinitely great ?

Ex. 3. Apply the inverse substitution, as in Ex. 2, to obtain the corresponding result
and the corresponding limit.

Ex. 4. Prove that a curve of finite length will meet an infinite number, or only a
finite number, of the circles Cn, according as it meets or does not meet the circle having
the line joining the common points of the substitution for diameter.

(Note. All these results are due to Poincare".)

It follows from what precedes that no real substitution can be loxodromic;
for, when the multiplier of a real substitution is not real, its modulus is
unity.

It is not difficult to prove that when a substitution, with complex
coefficients a, b, c, d, is parabolic, elliptic, or hyperbolic, then a + d is
either purely real or purely imaginary. In all other cases, the substitution
is loxodromic.

259.] ELLIPTIC SUBSTITUTIONS 521

Any loxodromic substitution can be expressed in the form

w — a z — a.
= ~

the coefficients of the quadratic determining a and ft are generally not real,
and the multiplier K, defined by

is a complex quantity such that, if

where p and co are real, then p is not equal to unity and <w is not zero.

260. Further, it is important to notice one property, possessed by elliptic
substitutions and not by those of the other classes: viz. an elliptic substitution
is either periodic or infinitesimal.

Any elliptic substitution of which a and @ are the distinct fixed points,
(they are conjugate imaginaries), can be put into the form

w — a. _ z — a.
wT^£ ~ tt 2—p >

where \K\ = I: \z\> K = eei. Then the rath power of the substitution is

_
wm - /3 z-/3

Now if 6 be commensurable with 2?r, so that

0/27T =

then, taking m = /*, we have

w^ — a _ z — a
w^$ ~ ~z^$

that is, WM = Zt

or the substitution is periodic.

But if 6 be not commensurable with 2?r, then, by proper choice of ra, the
argument m0 can be made to differ from an integral multiple of 2?r by a very
small quantity. For we expand <9/2?r as an infinite continued fraction : let
P/q, p'lq' be two consecutive convergents, so that p'q -pq'=±l. We have

where 77 < 1, that is, qQ - 2p7r = 2^77 - ,

522 INFINITESIMAL AND PERIODIC SUBSTITUTIONS [260.

where 77, being real, is numerically less than 1. Hence, taking m = q, we
have

wa - a z-a~_z-a.[^ 27777 .

wq-/3 z-ft z-PL q

where, by making q large, we can neglect all terms of the expansion after the
second. Then

(z - a) 0 - /3) 27T77 .

/).. /y X ' ^ J _. 1 A

-^s ijp*

that is, by taking a series of values of q sufficiently large, we can, for every
value of z find a value of w differing only by an infinitesimal amount from the
value of z. Such a substitution is called infinitesimal ; and thus the proposi
tion is established.

But no parabolic and no hyperbolic substitution is infinitesimal in the
sense of the definition. For in the case of a parabolic substitution we have

— OL Z — CL

which does not, by a proper choice of q, give wq nearly equal to z for every
value of z : and a parabolic substitution is not substitutionally periodic, that
is, it does not reproduce the variable after a certain number of applications.
But it may lead to periodic functions of variables : thus (z, z + o>) is a
parabolic substitution. And in the case of a hyperbolic substitution, we
have

where X is a real quantity which differs from 1. No value of q gives wq
nearly equal to z for every value of z : hence the substitution is not infini
tesimal. And it is not substitutionally periodic.

Similarly, a loxodromic substitution is not periodic, and is not infini
tesimal.

Hence it follows that, in dealing with groups of substitutions of the kind
above indicated, viz. discontinuous, all the elliptic transformations which occur
must be substitutionally periodic: for all other elliptic transformations are
infinitesimal. It is easy to see, from the above equations, that the effect of
an unlimited repetition of a parabolic substitution is to make the variable
ultimately coincide with the fixed point of the substitution ; and that the
effect of an unlimited repetition of a hyperbolic substitution is to make the
variable ultimately coincide with one of the fixed points of the substi
tution. These common points are called the essential singularities of the
respective substitutions.

261.] INVERSION CONNECTED .WITH SUBSTITUTIONS 523

261. It has been proved (§ 258) that a linear relation between two
variables can be geometrically represented as an inversion with regard to a
circle, followed by a reflexion at a straight line. The linear relation can be
associated with a double inversion by the following proposition*, due to
Poincare' : —

When the inverse of a point P with regard to a circle is inverted with
regard to another circle into a point Q, the complex variables of P and Q are
connected by a lineo-linear relation.

Let z be the variable of P, u that of its inverse with regard to the first
circle of centre/ and radius r; let w be the variable of Q, and let the second
circle have its centre at g and its radius s. Then, since inversion leaves the
vectorial angles unaltered, we have

(*-/)(*„-/„) = »-
for the first inversion, and

(w - g} (>0 - <70) = s2

for the second. From the former, it follows that

r~ s

and therefore --- - + — : -

z-f w—g

leading to w=

where, when «8 — /?7 = 1, we have

This proves the proposition.

Moreover, as the quantities /, g, r, s are limited by no relations, and as,
on account of the relation a.8 - /3y = 1, there are substantially only three
equations to determine them in terms of a, jB, 7, 8, it follows at once that the
lineo-linear relation can be obtained in an infinite number of ways by a pair of
inversions, and therefore in an infinite number of ways by an even number of
inversions.

Again, taking the two circles used in the above proof, we have

= (r± s)2 - d*,

Acta Math., t. iii, (1883), p. 51.

524 SUBSTITUTIONS AS INVERSIONS [261.

where d is the distance between the centres of the circles. Hence a + B
is real, and the substitution cannot be loxodromic. Moreover, if the circles
touch, the substitution is parabolic ; if they intersect, it is elliptic ; if they
do not intersect, it is hyperbolic.

Eliminating r and s between the equations which determine a, /3, 7, B, we
find

„ =

9

so that, when one centre is chosen arbitrarily, the other centre is connected
with it by the linear substitution*.

Ex. 1. Shew that, iff and g lie on the axis of real quantities, so that the substitution
is real, then

where X and p. are the fixed points of the substitution.

Hence prove that, if two real substitutions be given, it is generally possible to
determine three circles 1, 2, 3 such that the substitutions are equivalent to successive
inversions at 1 and 2 and at 1 and 3 respectively. Discuss the reality of these circles.

(Burnside.)

Ex. 2. Shew that, if a loxodromic substitution be represented in the preceding
geometrical manner, at least four inversions are necessary. (Burnside.)

This geometrical aspect of the lineo-linear relation as a double inversion
will be found convenient, when the relation is generalised from a connection
between the variables of two points in a plane into a connection between the
variables of two points in space.

* Burnside, Mess, of Math., vol. xx, (1891), pp. 163—166.

CHAPTER XX.

CONFORMAL REPRESENTATION: GENERAL THEORY.

262. IN Gauss's solution of the problem of the conformal representation
of surfaces, there is a want of determinateness. On the one hand, there is an
element arbitrary in character, viz., the form of the function ; on the other
hand, no limitation to any part of either surface, as an area to be represented,
has been assigned. And when, in particular, the solution is applied to two
planes, then, corresponding to any curve given in one of the planes, a curve
or curves in the other can be obtained, partially dependent on the form
of functional relation assumed, different curves being obtained for different
forms of functional relation.

But now a converse question suggests itself. Suppose a curve given in
the second plane : can a function be determined, so that this curve corresponds
to the given curve in the first plane and at the same time the conformal
similarity of the bounded areas is preserved, with unique correspondence
of points within the respective areas ? in fact, does the conformal corre
spondence of two arbitrarily assigned areas lead to conditions which can
be satisfied by the possibilities contained in the arbitrariness of a functional
relation ? And, if the solution be possible, how far is it determinate ?

An initial simplification can be made. If the areas in the planes,
conform ally similar, be T and R, and if there be an area 8 in a third plane
conformally similar to T, then 8 and R are also conformally similar to one
another, whatever S may be. Hence, choosing some form for 8, it will
be sufficient to investigate the question for T and that chosen form. The
simplest of closed curves is the circle, which will therefore be taken as 8 :
and the natural point within a circle to be taken as a point of reference is its
centre.

Two further limitations will be made. It will be assumed that the plane
surfaces are simply connected* and one-sheeted. And it will be assumed

The conformal representation of multiply connected plane surfaces is considered by
Schottky, Crelle, t. Ixxxiii, (1877), pp. 300—351.

526 RIEMANN'S THEOREM [262.

that the boundary of the area T is either an analytical curve * or is made up
of portions of a finite number of analytical curves — a limitation that arises in
connection with the proof of the existence-theorem. This limitation, initially
assumed by Schwarz in his early investigations •}• on conformal representation
of plane surfaces, is not necessary : and Schwarz himself has shewn J
that the problem can be solved when the boundary of the area T is any
closed convex curve in one sheet. The question is, however, sufficiently
general for our purpose in the form adopted.

Then, with these limitations and assumptions, Riemann's theorem^
on the conformation of a given curve with some other curve is effectively
as follows : —

Any simply connected part of a plane bounded by a curve T can always be
conformally represented on the area of a circle, the two areas having their
elements similar to one another ; the centre of the circle can be made the
homologue of any point 00 within T, and any point on the circumference of the
circle can be made the homologue of any point 0' on the boundary of T ; the
conformal representation is then uniquely and completely determinate.

263. We may evidently take the radius of the circle to be unity, for a
circle of any other radius can be obtained with similar properties merely by
constant magnification. Let w be the variable for the plane of the circle, z
the variable for the plane of the curve T] and let

log w= t = m + ni.

Evidently n will be determined by m (save as to an additive constant), for
m + ni is a function of z : and therefore we need only to consider m.

At the centre of the circle the modulus of w is zero, that is, em is zero :
hence m must be — oo for the centre of the circle, that is, for (say} Z = ZQ in T.

At the boundary of the circle the modulus of w is unity, that is, em is
unity: hence m must be 0 along the circumference of the circle, that is, along
the boundary of T.

Moreover, the correspondence of points is, by hypothesis, unique for the
areas considered : and therefore, as em and n are the polar coordinates of the
point in the copy and as m is entirely real, m is a one-valued function,
which within T is to be everywhere finite and continuous except only at
the point z0. Hence, so far as concerns m, the conditions are : —

(i) m must be the real part of some function of z :
(ii) m must be — oo at some arbitrary point z0 :

* A curve is said to be an analytical curve (§ 265) when the coordinates of any point on it
can be expressed as an analytical function (§ 3-1) of a real parameter,
t Crelle, t. Ixx, (1869), pp. 105—120.
J Ges. Werke, t. ii, pp. 108—132.
§ Ges. Werke, p. 40.

263.] ON CONFORMAL REPRESENTATION 527

(iii) m must be 0 along the boundary of T :

(iv) for all points, except z0, within T, m must be one-valued, finite and
continuous.

Now since m + ni = log w = log R + i®, the negatively infinite value of m
at zn arises through the logarithm of a vanishing quantity ; and therefore, in
the vicinity of z0> the condition (ii) will be satisfied by having some constant
multiple of log (z - z0) as the most important term in m + ni ; and the rest of
the expansion in the vicinity of z0 can be expressed in the form p(z — z0), an
integral rational series of positive powers of z — z0, because m is to be finite
and continuous. Hence, in the vicinity of z0, we have

log w = m + ni = - log (z -z0) + p(z- z0\

A,

where X is some constant. This includes the most general form : for the
form of any other function for m + ni is

- log {(z - z,} g(z- z0)} +P(z- z,},

where g is any function not vanishing when z = z0 : and this form is easily
expressed in the form adopted. Hence

Since w is one-valued, we must have X the reciprocal of an integer ; and
since the area bounded by T is simply connected and one-sheeted we must
have z - z0 a one-valued function of w. Hence X = 1 ; and therefore, in the
vicinity of z0,

w = (z-z0) &>&-**,

a form which is not necessarily valid beyond the immediate vicinity of z0,
for p (z - z0) might be a diverging series at the boundary. Thus, assuming
that p(z — ZQ) is 1 when z = z0, we have, in the immediate vicinity of z0,

m + ni = log (z - z0),
a form which satisfies the second of the above conditions.

It now appears that the quantity m must be determined by the con
ditions :

(i) it must be the real part of a function of z, that is, it must satisfy
the equation V2w = 0 :

(ii) along the boundary of the curve T, it must have the value zero :
(iii) at all points, except z0t in the area bounded by T, m must be
uniform, finite and continuous : and, for points z in the
immediate vicinity of z0, it must be of the form log r, where
r is the distance from z to z0.

528 RIEMANN'S THEOREM [263.

When m is obtained, subject to these conditions, the variable w is thence
determinate, being dependent on z in such a way as to make the area
bounded by T conformally represented on the circle in the w-plane.

264. The investigations, connected with the proof of the existence-
theorem, shewed that a function exists for any simply connected bounded
area, if it satisfy the conditions, (1) of acquiring assigned values along the
boundary, (2) of acquiring assigned infinities at specified points within the
area, (3) of being everywhere, except at these specified points, uniform, finite,
and continuous, together with its differential coefficients of the first and the
second order, (4) of satisfying V2u = 0 everywhere in the interior, except at
the infinities. Such a function is uniquely determinate.

But the preceding conditions assigned to m are precisely the conditions
which determine uniquely the existence of the function : hence the function
m exists and is uniquely determinate. And thence the function w is
determinate.

It thus appears that any simply connected bounded area can be conformally
represented on the area of a circle, with a unique correspondence of points in
the areas, so that the centre of the circle can be made the homologue of an
internal point of the bounded area.

An assumption was made, in passing from the equation

w = (z-Zo)eP{z-^

to the equation which determines the infinity of m, viz. that, when z = z0,
the value of p(z— z0) is 1. If the value of p(z— z0) when z=z0 be some
other constant, then there is no substantial change in the conditions:
instead of having the infinity of m actually equal to log z — z0 , the new
condition is that m is infinite in the same way as log z — z0 , and then a
constant factor must be associated with w. A constant factor may also arise
through the circumstance that n is determined by m, save as to an additive
constant, say 7 : hence the form of w = em+ni will be

w = A'eyiu = Au.

Since displacement in the plane makes no essential change, we may take
a form w — Au + B, where now the conformal transformation given by w is
over any circle in its plane, the one given by u being over a particular circle,
centre the origin and radius unity.

The conformation for w is derived from that for u by three operations :
(i) displacement of the origin to the point — Bj A :
(ii) magnification equal to A' :
(iii) rotation of the circle round its centre through an angle 7 :

264.] DERIVATIVE FUNCTIONS REQUIRED 529

these operations evidently make no essential change in the conformation.
If the limitation to the particular circle, centre the origin and radius 1,
be made, evidently B = 0, A' =•!, but 7 is left arbitrary. This constant
can be determined by assigning a condition that, as the curve G has its
homologue in the circle, one particular point of C has one particular point of
the circumference for its homologue : the equation of transformation is then
completely determined.

This determination of A', B, 7 is a determination by very special con
ditions, which are not of the essence of the conformal representation : and
therefore the apparent generality for the present case should arise in the
analysis. Now, if w = Au + B, we have

d f, fdw\] d f, fdu^

which is the same for the two forms ; and therefore the function to be

sought is

d (.
log

dz\ &\dz,

when the area included by C is to be represented on a circle so that a given
point internal to C shall have the centre of the circle as its homologue.
The arbitrary constants, that arise when w is thence determined, are given
by special conditions as above.

Again, if the conformation be merely desired as a representation of the
2-area bounded by the analytical curve G on the area of a circle in the
w-plane (without the specification of an internal point being the homologue
of the centre), there will be a further apparent generality in the form of the
function. From what was proved in § 258, a circle in the w-plane is trans
formed into a circle in the w-plane by a substitution of the form

_Au + B

~Cu^~D'

so that, if w be a special function, w will be the more general function giving
a desired conformal representation ; and, without loss of this generality, we
may assume AD — BC = 1. Using {w, z} to denote

d2 I , dw\ , f d

that is,

" \w

called the Schwarzian derivative by Cayley*, we have

{w, z} = {u, z},

* Camb. Phil. Trans., vol. xiii, (1879), p. 5; for its properties, see Cayley's memoir just quoted,
pp. 8, 9, and my Treatise on Differential Equations, pp. 92, 93.

F. 34

530 SOLUTION BY BELTRAMI AND CAYLEY [264.

which is the same for the two forms: and therefore the function to be

sought is

{w, z],

when the area included by the analytical curve C is to be conformally repre
sented on a circle. The (three) arbitrary constants, that arise when w is
thence determined, are obtained by special conditions.

These two remarks will be useful when the transforming equation is
being derived for particular cases, because they indicate the character of the
initial equation to be obtained : but the importance of the investigation is
the general inference that the conformal representation of an area bounded
by an analytical curve on the area of a circle is possible, though, as the proof
depends on the existence-theorem, no indication is given of the form of
the function that secures the representation.

Further, it may be remarked that it is often convenient to represent a
2-area on a w-half-plane instead of on a w-circle as the space of reference.
This is, of course, justifiable, because there is an equation of unique transfor
mation between the circular area and the half-plane ; it has been given (Ex. 9,

, au + b . ,.,, .,, f
S 257). Moreover, a further change, given by u = — — --. , is still possible: tor,

•^ CU ~\~ CL

when a, b, c, d are real, this transformation changes the half-plane into itself,
and these real constants can be obtained by making points p, q, r on the
axis change into three points, say 0, 1, oo , respectively — the transformation

then being

, u — p q — r
u = - - .

u—r q—p

265. Before discussing the particular forms just indicated, we shall
indicate a method for the derivation of a relation that secures conformal
representation of an area bounded by a given curve C.

Let* the curve C be an analytical curve, in the sense that the coordinates
x and y can be expressed as functions of a real parameter, say of u, so that
we have x = p (u), y = q (u) ; then

z = a; + iy =p + iq = <j> (u)-
If for u we substitute w = u + iv, we have

z = (f> (w) ;

and the curve C is described by z, when w moves along the axis of real
quantities in its plane.

When the equation x + iy= <f> (u + iv) is resolved into two equations
involving real quantities only, of the form x = X (u, v), y = p (u, v), then the
eliminations of v and of u respectively lead to curves of the form
i/r (ar, y, u) = 0, % (x, y, v) = 0,

* Beltrami, Ann. di Mat., 2da Ser., t. i, (1867), pp. 329—366; Cayley, Quart. Journ. Math.,
vol. xxv, (1891), pp. 203—226; Schwarz, Ges. Werke, t. ii, p. 150.

265.] FOR ANALYTICAL CURVES 531

which are orthogonal trajectories of one another when u and v are treated as
parameters. Evidently % (x, y, 0) = 0 is the equation of G : also

So far as the representation of the area bounded by C on a half-plane is
concerned, we can replace w by an arbitrary function of Z(= X + iY) with
real coefficients: for then, when Y — 0, we have w =f(X) and

which lead to the equation of G as before, for all values of f. This arbi
trariness in character is merely a repetition of the arbitrariness left in
Gauss's solution of the original problem.

Now let the w-plane be divided into infinitesimal squares with sides
parallel and perpendicular to the axis of real quantities. Then the area
bounded by G is similarly divided, though, as the magnification is not every
where the same, the squares into which the area is divided are not equal to
one another. The successive lines parallel to the axis of u are homologous
with successive curves in the area, the one nearest to that axis being the
curve consecutive to G. Similarly, if the ^-plane be divided.

Conversely, if a curve consecutive to G, say G', be arbitrarily chosen, then
the space of infinitesimal breadth between G and G' can be divided up into
infinitesimal squares. Suppose the normal to G at a point L meet G' in L' :
along G take LM = LL', and let the normal to G at M meet G' in M' ; along
G take M N = M M ', and let the normal to G at N meet G' in N' : and so on.
Proceeding from C" with L'M', M'N', ... as sides of infinitesimal squares, we
can obtain the next consecutive curve G", and so on ; the whole area bounded
by G may then be divided up into an infinitude of squares. It thus appears
that the arbitrary choice of a curve consecutive to G completely determines
the division of the whole area into infinitesimal squares, that is, it is a
geometrical equivalent of the analytical assumption of a functional form
which, once made, determines the whole division.

Next, we shall shew how the form / of the function can be determined
so as to make the curve consecutive to G a given curve. As above, the
curve G is given by the elimination of a (real) parameter between

as=p(u), y = q(u);
and the representation is obtained by taking

x + iy = z=p (W) + iq (W) =p [f(Z)}+iq (f(Z)}.

Let the arbitrarily assumed curve C', consecutive to C, be given by the
elimination of a (real) parameter 6 between

where p, P, q, Q are functions with real coefficients, and e is an infinitesimal

34—2

532 CONFORMATION OF AREA [265.

constant : the form of/ has to be determined so that the curve corresponding
to an infinitesimal value of Y is the curve (7. Taking u=f(X), where
u and X are real, we have, for the infinitesimal value of Y,

v du , vdu ,

so that ac=p—Yjv(l> y = cL + dX?'

dashes denoting differentiation with regard to u. This is to be the same
as the curve C', given by the equations

x = p + eP, y = q + eQ.

Hence the (real) parameter 6 in the latter differs from u only by an infini
tesimal quantity : let it be u — p, so that we have

x=p-pp' + eP, y = q - pq' + eQ,

the terms involving products of e and fj, being neglected, because they are of
at least the second order. Hence

/ -n TT- CL1H r t r\ TT- vt-w /

-/*/+ eP = - Y-^q, -M'+eQ = ? dXP 5

whence /* (p* + q2) = e (Ppr + Qq'),

and e (p'Q - q'P} = Y || (p* + q'2) *

Now e is a real infinitesimal constant, as is also Y for the present purpose :
so that we may take e = AY, where A is a finite real constant: and A may
have any value assigned to it, because variations in the assumed value
merely correspond to constant magnification of the ^-plane, which makes no
difference to the division of the area bounded by C. Thus

fit/

'

_

and therefore A X = I —^ — ^=i du,

) p'Q-qP

the inversion of which gives u =f(X) and therefore w —f(Z), the form
required.

AI AvpP' + Q<l'

Also we have //, = A Y -~ - ,~~ ,

shewing that, if the point x = p + eP, y = q + eQon C' lie on the normal to G
at x=p, y = q, the parameters in the two pairs of equations are the same;
the more general case is, of course, that in which the typical point on C' is in

* Beltrami obtains this result more directly from the geometry by assigning as a condition
that the normal distance between the curves is equal to the arc given by du : I.e., (p. 530, note),
p. 343.

265.] BOUNDED BY ANALYTICAL CURVE 533

the vicinity of C. And it is easy to prove that the normal distance between
the curves at the point in consideration is

Ff*l

dX'

where ds is an arc measured along the curve C.

Ex. 1. As an illustration*, let C be an ellipse x2/a?+y2/b* = l and let C' be an interior
confocal ellipse of semi-axes a - a, b - ft, where a and ft are infiiiitesimally small ; so that,

since

(a-a)2-(&-/3)2=«2-&2=c2,

ft ft

we have aa = 6/3 = ce say; then the semi-axes of C" are a — f, b — jt. We have

Ct 0

p = a cos u, q = b sin u,
P=--cosu, Q=-jsinu,

. fab , ab

so that A X = \ — du = — u,

J c c

or, taking .4 = —, we have X=u and therefore Z=w. Hence the equation of transfer-

C

mation is

Z+ibsmZ ;

or, if a = ccosh F0, 6=csinh Y0, and if Y' denote Y0- Y, the equation is

z = c cos (X+ i Y') = c cos Z' .

The curves, corresponding to parallels to the axes, are the double system of confocal
conies.

Ex. 2. When the curve C is a parabola, with the origin as focus and the axis of real
quantities as its axis, and C" is an external confocal coaxial parabola, the relation is

substantially the same relation as in Ex. 7, § 257.

Ex. 3. When G is a circle with its centre on the axis of real quantities and C' is
an interior circle, having its centre also on the axis but not coinciding with that of C, the
circles being such that the axis of imaginary quantities is their radical axis, the relation

can be taken in the form

z = ctanZ. (Beltrami ; Cayley.)

Note. Although, in the examples just considered, the successive curves C
ultimately converge to a curve of zero area (either a point or a line), so that
the whole of the included area is transformed, yet this convergence is not
always a possibility, when a consecutive to C is assigned arbitrarily. There
will then be a limit to the ultimate curve of the series, so that the repre
sentation ceases to be effective beyond that limit. The limitation may

cl 2

arise, either through the occurrence of zero or of infinite values of -y^. for

dZJ

areas and not merely for isolated points, or through the occurrence of
branch-points for the transforming function. In either case, the uniqueness
of the representation ceases.

* Beltrami, I.e., (p. 530, note), p. 344 ; Cayley, (ib.), p. 206.

534 EXAMPLES [265.

Ex. 4. Consider the area, bounded by the cardioid

r = 2a(l+cosd) ;
then we can take

x=p = %a (1 +cos u) cos u, y — q = ^a (1 + cos u)$ir\ u,
where evidently u=6 along the curve. Let the consecutive curve be given by

x= — ae + 2a(l + e) (1 +cos u'} cos u', y = 2a (1 +e) (1 + COSM') sinu',
so that, to determine X, we assume P= — a + 2«(l+cos «)cosw, Q = 2a (1+cos u) sin u,
for u' — u=—fj. a small quantity.

We have p'2 + q'2 = 1 6a2 cos2 %u,

p'P + q'Q — — 2a2 sin u ;

and then, proceeding as before and choosing A of the text as equal to - 1, (which implies
that e is negative and therefore that the interior area is taken), we find

X=u,

therefore Z=w. Thus the cardioid itself and the consecutive curves are given by

z = p + iq = 2a (1 + cos Z} eiz.
To trace the curves, corresponding to lines parallel to the axes of X and F, we have

Hence, multiplying, we have

r = 4a<

= 2ae~ Y (cosh F+ cos A") ;
and, dividing, we have

r = 4ae ~ Y (cos ^f

COS

, . i(jf

that is, 6

_

6 -- T~~T? i 1^1? -~- i~v • T i^r>-»

cos f A cosh f / - « sin ^ A sinh £ r

and therefore tan \( X — 6} = tan ^X tanh | F.

Moreover, we have -.

which vanishes when ^— TT (2w + l), that is, at the point X=(2n + l) ir, F=0 ; whence the
cusp of the cardioid is a singularity in the representation.

When F=0, then X—6 and r = 2a (1 + cos^), which is the cardioid ; when F is very
small and is expressed in circular measure, then

tan J( X - 6) - £ F tan \X,

or AT=0+Ftan£0,

so that r = 2a (1 + cos 6} - 4aF.

It is easy to verify that Q=u' + jFtan £M',

agreeing with the former result.

The relation may be taken in the form

which shews that z = a is a branch-point for Z. Two different paths from any point to a

265.]

OF ANALYTICAL CURVES

535

point P, which together enclose a, give different values of Z at P. Hence the representa
tion ceases to be effective for any area that includes the point a.

Consider a strip of the Z-plane between the lines F=0, Y= + oo , X— -|TT, X= + ^TT.
First, when Z=^ir+iY, we have X=^rr, so that

and therefore
whence

tan %6 = e~ * ,

2a

r =

— - - ,
1 + cos 6

a part of a parabola. And when Y varies from QO to 0, 6 varies from 0 to \n.
Secondly, when Z=X, so that Y=0, we have X=6, and then

r = 2a(l+cos<9):

and, when X varies from £TT to -\ir, 6 varies from JTT to -|?r.
Thirdly, when Z= -far+iY, we have X= -^TT, so that

tan ( JTT + %6] = tanh £ Y,

whence tan \6 = -e~Y,

so that, as Y varies from 0 to oo , Q varies from - \n to 0. And then

~1+COS0'

another part of the same parabola as before.

Lastly, when Y is infinite and X varies from - - to + - , we have

so that 0=0; and then r = a, in effect the point of the 2-plane corresponding to the
point at infinity in the Z-plane.

We thus obtain a figure in the 2-plane ABCDA corresponding to the strip in the
Z-plane : the boundary is partly a parabola DAB, of focus 0 and axis OA, and partly
a cardioid with 0 for cusp — the inverse of the parabola with regard to a circle on the
latus rectum BD as diameter : the angles at B and D are right.

X=-iir

Fig. 92.

To trace the division of the space between the axes of the cardioid and of the parabola
corresponding to the division of the plane strip into small squares, we can proceed as
follows.

536 EXAMPLES [265.

Let e~ =c : then we have

- =cf- + c) + 2c cos X,
a \c J

or, if R = ap, then p = l+c2 + 2ccos X;

and tan A (X - 6} = = — tan ^X,

i + c

so that

- — . . •
sm(X-^d)

, ., , eosi# sinA0 1

and therefore — • — ^= — ^-^=-7-,

1+ccosA csmJT *Jp

so that c cos X= Vp cos \8 - 1 , c sin X=>Jp sin |0,

from which the curves, corresponding to c = constant and to X= constant, are at once
obtained. They are exhibited in the figure, the whole of the internal space being
divisible.

By combination with the transformation, which (Ex. 12, § 257) represents a strip of
the foregoing kind on a circle, the relation can be obtained, leading to the representation
of the figure on a circle.

Ex. 5. Shew that, if a straight line be drawn from the cusp to the point r = a, 0 = 0, so
as to prevent z from passing round 2 = 0 or z = a, then the area bounded by the cardioid
and this line can be represented, on a strip of the w-plane given by Y— 0, Y= oo ,
X= — TT, X= +TT, by the equation

ff - 1} . (Burnside.)

Ex. 6. In the same way, treating the curve (the Cissoid of Diocles)
and taking the equations

sin3w

as defining the points on the curve, we may assume the consecutive curve defined by the
equations

another cissoid with the same asymptote. Proceeding as before we find the value of X
to be tan u + jV tan3 u, on taking A = — §r.

The relation, which changes the cissoidal arc into the axis of X and a consecutive
cissoidal arc into a line parallel to the axis of X at an infinitesimal distance from it,
is then

. sin2 w Wi
z = 2r~ — e™,
cosw

where the relation between w and Z is

Z= tan

Note. The method is applicable to any curve, whose equation can be expressed in the
form r=f(6) : a first transformation is

z=f(w)ewi.

The determination of w in terms of Z depends upon the character of the consecutive
curve chosen ; this curve also determines the details of the conformation.

266.] CONTRACTION OF AREAS 537

266. It has been pointed out (§ 265, Note) that, though a curve and its
consecutive in the ^-plane correspond with a curve and its consecutive in the
w-plane, the conformation is only effective for parts of the included areas,
in which the magnification, if it is not uniform, becomes zero or infinite only
at isolated points, and in which no branch-points of the transforming relation
occur. The immediate vicinity of a curve C is conformable with the
immediate vicinity of a corresponding curve $, arbitrarily chosen limits
being assigned for the vicinity.

But, as remarked by Cay ley*, when a curve is given, then the con
secutive curve can be so chosen that the whole included area is conformable
with the whole corresponding area in the ^-plane. For a circle can be thus
represented, the ultimate limit of the squares when consecutive curves are
constructed being then a point : this can be expressed by saying that the
area can be contracted into a point. For instance, the relation

z(w + l) + i(w- 1)= 0

transforms the ^-half-plane into the area included by a w-circle of radius
unity. The lines parallel to the axis of x are internal circles all touching
one another at the point (—1, 0) : and the lines parallel to the axis of y are
circles orthogonal to these, having their centres on a line parallel to the axis
of Fand all touching at the point (—1, 0). Similarly for the contraction of
any circle, by making it one of two systems of orthogonal circles : the form of
the necessary equation is obtained as above by taking the next circle of the
same system as the consecutive curve : and a circle can thus be contracted to
its centre (the infinitesimal squares being bounded by concentric circles and
by radii) when the w-circle is derived from a strip of the ^-half-plane by the
relation w = eiz. Such a contraction of a circle is unique.

But, by Riemann's theorem, it is known that the area of a given analy
tical curve can be conformally represented on the area of a given circle, so
that a given internal point is the homologue of the centre and a given point
on the curve is the homologue of a given point on the circumference of the
circle : and that the representation is unique. Hence it follows that, when
an analytical curve C is given, a consecutive curve G' can be chosen in such a
manner as to secure that the construction of the whole series of consecutive
curves by infinitesimal squares will make the curve C contract into an
assigned point -f*.

267. The areas, already considered in special examples, have been
bounded by one or by two analytical curves : we shall now consider two
special forms of areas bounded by a number of portions of analytical curves.
These areas are (i) the area included within a convex rectilinear polygon,
(ii) the area bounded by any number of circular arcs, and especially the area

* I.e. (p. 530, note), pp. 213, 214.

t For further developments, see Cayley's memoir cited p. 530, note.

538 RECTILINEAR POLYGON [267.

bounded by three circular arcs. For the sake of analytical simplicity, the
former will be conformally represented on the half-plane, the transformation
to the circle being immediate by means of the results of § 257.

In regard to the representation* of the rectilinear polygon, convex in
the sense that its sides do not cross, we shall take the case corresponding
to the first of the two forms of § 264 ; it will be assumed that the origin in
the w- plane is left unspecified and that the magnification is subject to an
unspecified increase, constant over the plane. Our purpose, therefore, is to
represent the w-area included by a polygon on the half of the z-plane ; the
boundary of the polygonal area in the w-plane is to be transformed into the
axis of real quantities in the 2-plane.

It follows from Schwarz's continuation-theorem (§ 36), that a function
defined for a region in the positive half of a plane and acquiring continuous
real values for continuous real values of the argument can be continued across
the axis of real quantities: and the continuation is such that conjugate
values of the function correspond to conjugate values of the variable. More
over, the function, for real values of the variable, can be expanded in a
converging series of powers, so that

w_— w0 = (x — c)P (x — c),

where P is a series of positive, integral powers with real coefficients that does
not vanish when c is the value of the real variable x.

Suppose a convex polygon given in the w-plane, the area included by
which is to be represented on the £-plarie, and the contour of which is to be
represented along the axis of x by means of a relation between w and z.

First, consider a point say /3 on the side Ar^Ar which is not an angular
point. Then, if 6 denote the inclination of Ar_^Ar to
the axis of u, the function

(w - 13) <r *<-+*>

is real when w lies on the side Ar_iAr: it changes sign

when w passes through ft : and for all other points w,

lying either in the interior or on the other sides of the -

polygon, it has the same properties as w. Hence, if b be /

a (purely real) value of z corresponding to w = ft, we have r~1 Fig 93<

(w - /3) e-^+o* = 0 - 6) P 0 - b\

* In connection with the succeeding investigations the following authorities may be
consulted :

Schwarz, Ges. Werke, t. ii, pp. 65—83 ; Christoffel, Ann. di Mat., 2da Ser., t. i, (1867),
pp. 95—103, ib., t. iv, (1871), pp. 1—9; Schlafli, Crelle, t. Ixxviii, (1873), pp. 63—80;
Darboux, Theorie generate des surfaces, t. i, pp. 176—180 ; PhragmSn, Acta Math., t. xiv,
(1890), pp. 229—231.

267.] REPRESENTED ON A HALF-PLANE 539

for points in the vicinity of (3 : the series P(z-b) does not vanish for z = b ;
and, when w lies on the side ArAr^, then z — x.

Next, consider the vicinity of an angular point of the polygon. Let 7 be
the coordinate of Ar, let pir be the internal angle of the polygon, and let
the inclination of ArAr+l to the axis of u: and consider the function

When w lies on the side ArAr_^ at a distance d from Ar, then

w - 7 = rfe»>+«) ,
so that the function is then real and positive.

When w lies in the interior of the polygon, the function has the same
properties as w, and its argument is negative.

When w lies on the side ArAr+1 at a distance d' from Ar, then w — 7 = d'e1*,
so that the function is d/e-l'(ir+*-*', that is, d'e-***. Hence

i

{(w-7)e-l>+e>JM

is real and positive along the side Ar_^Ar, and is real and negative along
the side ArAr+1. If then z = c be the value corresponding to w = y, we
can expand this function in the form (z - c) Q' (z — c): and therefore

(w - 7) e-icH-*) =(e^ cy R(z- c),
where R (= Q/(X) does not vanish for z = c.

These forms assume that neither b nor c is infinite. The point on the
boundary of the polygon (if there be one), corresponding to x — oo , can be
obtained as follows. We form a new representation of the ^-plane given by

*£=-!,

which conformally represents the upper half of the 0-plane on itself: and
then, on the assumption that such point at infinity does not correspond
to an angular point of the polygon, we have £ = 0 corresponding to an
ordinary point of the boundary, so that

where Q does not vanish when z= oo .

All kinds of points on the boundary of the w-polygon have been considered,
corresponding to points on the axis of x.

We now consider points in the interior. If w' be such an interior point
and z' be the corresponding z-point, then

w-w' = (z-z')S(z-z'\

where 8 does not vanish for z = z* because at every point .. must be different

dLz

540 RECTILINEAR POLYGON [267.

from zero : for otherwise the magnification from a part of the z-plane to a
part in the interior of the polygon would be zero and the representation
would be ineffective.

Now in the present case, just as in the first case suggested in § 2G4, it is
manifest that, if a particular function u give a required representation, then
Au + B, where \A = 1, will give the same w- polygon displaced to a new
origin and turned through an angle = arg. A, that is, no change will be made
in the size or in the shape of the polygon, its position and orientation in the
w-plane not being essential. Hence the function to be obtained may be
expected to occur in the form w = Au +B, so that, in representing a figure
bounded by straight lines, the function to be obtained is

„ d (, fdw

Z = -j- { log -7-
dz\ ° \dz

Now in the vicinity of a boundary-point /3, not being an angular point
and corresponding to a finite value of z, we have

w - & = e^+V (z-b}P(z- b),
and therefore Z = P1 (z — b),

having z = b for an ordinary (non-zero) point.

For a boundary-point /3', not being an angular point and corresponding to
an infinite value of z on the real axis, we have

z z

and therefore Z = h - Qi ( -

z z2- \z

where Q1 is finite for z = oo . Thus Z vanishes for such a point.
In the vicinity of an angular point 7, we have

w-y = e^+e> (z - cY R(z- c),

and therefore Z = — — + R,(z — c),

z — c

where Rt has z = c for an ordinary point.

Lastly, for a point w' in the interior of the polygon, we have

w-w' = (z-z'}8(z-z'},
and therefore Z = $t (z — z'},

having z = z' for an ordinary point.

Hence Z, considered as a function of z, has the following properties : —

It is an analytical function of z, real for all real values of its argument,
and zero when x is infinite :

267.] REPRESENTED ON A HALF-PLANE 541

It has a finite number of accidental singularities each of the first order
and all of them isolated points on the axis of x : and at all other
points on one side of the plane it is uniform, finite and continuous,
having (except at the singularities) real continuous values for real
continuous values of its argument.

The function Z can therefore be continued across the axis of x, conjugate
values of the function corresponding to conjugate values of the variable : and
its properties make it, by § 48, a rational, algebraical, meromorphic function
of z.

Let a, b, c, ..., I be the points (all in the finite part of the plane) on the
axis of x corresponding to the angular points of the polygon, and let

cor, PTT, 773-, ..., XTT

be the internal angles of the polygon at the respective points : then (by § 48)
^o-^/5-l +X-1
z — a z — o z — l

no additive constant being required because Z has been proved to vanish for
infinite values of z.

Moreover, because OLTT, fin, ..., \TT are the internal angles of the polygon,
we have

S (TT — «TT) = 2?r,

so that 2 (a - 1) = - 2,

a relation among the constants a, /3, ..., A, in the equation
d f, fdw\) a - 1 X - 1

and each of the quantities a, /3, ..., A. is less than 2. This equation*, when
integrated, gives

w = Cf(z- a)"-1 (z - b)^ ...(z- If-* dz + C',

where C and C' are arbitrary constants, determinable from the position of the
polygon f.

268. It may be remarked, first, that any three of the real quantities
a, b, c, ..., I can be chosen arbitrarily, subject to the restrictions that the
points a, b,c,...,l follow in the same order along the axis of x as the angular
points of the polygon and that no one of the remaining points passes to
infinity. For if three definite points, say a, b, c, have been chosen, they can,
by a real substitution

* This relation, as is possible with many relations in conformal representation of areas, is
made the basis of some interesting applications in hydrodynamics, by Michell, Phil. Trans., (1890),
pp. 389—431 ; and in conduction of heat, by Christoffel, I.e., p. 538, note.

t This result was obtained independently by Christoffel and by Schwarz : I.e., p. 538, note.

542 RECTILINEAR POLYGON [268.

where p, q, r, s are real quantities satisfying ps — qr — 1, be changed into
other three, say a, b', c' : and then, substituting

and using the relation 2 (« — 1) = — 2,

we have w — r/(£— a')*"1 (f— 6')^~1 ... (£ — Z')v~1rf£' + (7',

where F is a new constant. By the real substitution, the axis of real

quantities is preserved : and thus the new form equally effects the conforrnal

representation of the polygon.

But, secondly, it is to be remarked that when three of the points on the
axis of x are thus chosen, the remainder are then determinate in terms of
them and of the constants of the polygon.

Note. The £-point at infinity has been excluded from being the homo-
logue of one of the angular points of the w-polygon : but the exclusion is not
necessary.

If z = co be the homologue of an angular point cr, at which the internal
angle is /U,TT, then proceeding as before, we have

(w - 7) e-'>+9> =-.

for points in the vicinity of a ; and therefore

d (, fdw \\ u + 1 11

-j- 4 log I ~=- I r = h terms in - , - ,

Let a, b, c, . . . , k be the homologues of the other vertices where the angles
are «TT, /3-Tr, ..., KTT: then the function

d L /dw\\ a- 1 /3-1 «-l

' — \ lOff I I r •

dz\ h\dzj) z—a z — b z — k

is finite at a, b, ..., k. The term in - in the fractional part is

But fjb — l + S(a — 1) = — 2, so that the term is — -- . Hence the function

z

for infinite values of z begins with - , and therefore it vanishes at that point.

z~

It has thus no infinities for any value of z : being a uniform function, it is
therefore a constant, which (owing to the value of the function for z = co ) is
evidently zero : so that

d L fdw\] a- 1 13-1 K-l

Hence, if one of the angular points of the polygon be made to correspond

268.] TRIANGLE ON A HALF-PLANE 543

to an infinite value of z, the equation which determines the conformal
representation is

w = Aj(z- a}*-1 (z - by~l ...(z- ky~l dz + B,
where a-

fjLTT (usually equal to zero) being the internal angle at the vertex which has
its homologue at infinity.

269. The simplest example is that of a triangle of angles air, fiir, 773% so
that

Then a particular function determining the conformal representation of this
w-triangle on the half 2-plane is

so that

f dz

ij — I _ _ ______ ____ ___ _

J (z - ay-- (z - by-f*~(z -cy-y '

dz

a differential equation of the class partially discussed in §§ 246 — 252.

For general values of a, ft, 7 the integral-function tv is an Abelian
transcendent of some class which is greater than 1 : and then, after §§ 110,
239, z is no longer a definite function of w, and the path of integration must
be specified for complete definition of the function.

If a = 0, the only instance when the integral is a uniform function of w
is when # = £, 7 = ^: arid then the function is singly-periodic (§ 252, III.).
In such a case the w-figure is a strip of the plane of finite breadth, extending
in one direction to infinity and terminated in the finite part of the plane by
a straight line perpendicular to the direction of infinite extension.

If no one of the quantities a, /3, 7 be zero, then on account of the condition
a + ft + 7 = 1, the only cases when the integral gives z as a uniform function
of w are as follows. In each case the function is doubly-periodic.

(§ 252, III, 10). . .(A) : «=£, 0 = £, 7 = $ : an equilateral triangle.

(ib., 9). . .(B) : a = ^, /3 = £, 7 = £ : an isosceles right-angled triangle.

(ib., 8)... (C): a=^,/3 = 1, 7 = J : a right-angled triangle with one

angle equal to ^TT.

The integral expressions for these cases have been given by Love *, who has
also discussed a further case, (due to Schwarz, Ex. 3, § 252), in which z occurs
as a two-valued doubly-periodic function of w ; the triangle is then isosceles
with an angle of |TT, the values of a, /3, 7 being a = f , /9 = £, 7 = £.

* Amer. Journ. of Math., vol. xi, (1889), pp. 158—171.

544 SQUARE ON A CIRCLE [269.

The example next in point of simplicity is furnished by a quadrilateral,
in particular by a rectangle : then

a=/3 = 7=S = i:
and the general form is

w=f{(z-a)(z- b) (z -c)(z- d)}-± dz,
so that z is a doubly-periodic function of w.

First, let it be a square : and choose oo , 1, 0 as points on the axis of x
corresponding to three of the angular points in order. The symmetry of the
w-figure then enables us to choose — 1 as the remaining angular point.

In the vicinity of z = K, we have

z — K

a finite quantity, where K — 0, 1, — 1 in turn.
For infinite values of zt we have

where T is finite for z = oo : hence

dz

Hence the function
d

, /dw\] 1 .11

log T~ r = — 9 + terms m -,-,....
& \dz)} * z z> z3

is finite for z = 0, z = 1, z = — I : it is zero for 2= oo : it is not infinite for
any other point in the plane. It is a uniform function of z : it is therefore a
constant, equal to its value at any point, say, at z = oo where it is zero : and so

d f, /cfaA) . / 1 1 1

H r

U + i ^ -^-i/'

/"2 ,

whence w =

(7 and (7' being dependent upon the position and the magnitude of the
^-square.

Again, the half .z-plane is transformed into the interior of a ^-circle, of
radius 1 and centre the origin, by the relation

1 + 2

Then except as to a constant factor, which can be absorbed in C, the integral

in w changes to

dZ

[ di
J(T^

269.]

RECTANGLE ON A HALF-PLANE

545

so that, by the relation

W =

dZ

o (!-£*)*'

the interior of a ^-circle, centre the origin and radius 1, is the conformal
representation of the interior of some
square in the TT-plane. Denoting by

ri fix

L the integral I 7= ^, so that 2L

J oCl-O4

is the length of a diagonal, the angular

points of the square are D, A, B, C

on the axes of reference : and these

become d, a, b, c on the circumference

of the circle. They correspond to - 1, 0, 1, oo on the axis of as in the

representation on the half-plane.

Ex. Shew that the area outside a square in the w-plane can be conformally repre
sented on the interior of a circle in the 2-plane, centre the origin and radius unity, by the
equation

»-/* 3 (!+**)*<&,

Fig. 94.

the 2-origin corresponding to the infinitely distant part of the w-plane. (Schwarz.)

Secondly, let the rectangle have unequal sides. Then the symmetry of

the figure justifies the choice of y , 1, - 1, - y as four points on the axis of x

corresponding to the angular points of the rectangle when it is represented
on the half-plane. We thus have

w = C \ {(1 - z-) (1 - fcV))-* dz + C'.
J o

the rectangle be taken so that its angular points are a, a + 2bi, —a + 2bi,
a in order, these corresponding to 1, y , — T , — 1 respectively, then we have

/C Iv

so that the relation is

and then

a = CK,

K

26

Znb

whence q = e a ,

where q is the usual Jacobian constant : this equation determines the relation
between the shape of the rectangle and the magnitude of k.

F. 35

546 QUADRILATERAL [269.

In the particular case when the rectangle is a square, we have b = a and
so q = e~2n, or T> = 2 : and therefore* k = 3 - V8 or j- = 3 + V8. The differ
ence from the preceding representation of the square is that, there, the point
z = i was the homologue of the centre of the square, whereas now, as may
easily be proved, the point z = i (V2 + 1) is the homologue of the centre.

But in the case of a quadrilateral in which such symmetrical forms are
obviously not possible and, in the case of any convex polygon, only three points
can be taken arbitrarily on the axis of x : the most natural three points to take
are 0, 1, oo for three successive points. The values for the remaining points
must be determined before the representation can be considered definite.

Thus in the case of a quadrilateral, taking GO , 0, 1 as the homologues of

D. A, B respectively and - as the homologue of C, ^^

P D --^^^7r

(where /JL< 1), the equation for conformal representation

is

w = Cu + C',

where lan

FZ p A

u = za~^ (1 -zf-1 (1 - f**)*"1 d* = Adz, say. Fig. 95.

Jo •* °

If the w-origin be taken at A, and the real axis along AB, we have

a = C I l Xdx + C',

o

de*™ = c Xdx + C',
o

i

r
J i

Xdx + C',

being the equations for the four angular points. They determine only three
quantities G, C', ^ so that they coexist in virtue of a relation, which is in
effect the relation between the sides and the angles of a quadrilateral.
An equation to determine p is

Too F\

a Xdx = deina I Xdx;
Jo Jo

the second equation serves to determine C, because C' = 0.

The equation determining p can be modified as follows^, so as to be expressed
in terms of the hypergeornetric series.

* This is derived at once by means of the quadric transformation in elliptic functions.

t For the analytical relations in reference to the definite integrals, see Goursat, "Sur
1'equation differentielle liniSaire cmi admet pour integrale la serie hypergeometrique," Ann.
de VEc. Norm. Sup., 2™ Ser., t. x, (1881), Suppl., pp. 3—142 ; and for the relations between the
hypergeornetric series, see my Treatise on Differential Equations, pp. 192—201, 232, 233, the
notation of which is here adopted.

269.] DETERMINATION OF CONSTANT 547

Let - etrra = '\, so that the equation is
ct

r xdx=\ r

Jo Jo

Xdx.

Now to compare these integrals with the definite integrals which are the solution
the differential equation of the hypergeometric series, we take

so that
And

so that, as p.<l, the definite integral is finite at all the critical points.
We have

r(/3')r(y-/3') „, , ,

= — --- "

r(a+/3) »

-pi i-p, y -«'-#'+!, ^—

/*'-y'+l, !-„', 2-y',

"(y+«) 2'

Hence (X- 1)

NOW if M= -

- '

- -

n (i - y') n (y - a' - 1) n (y - /3' - 1) r (y +S) r (1 - 8) r OS) '

= n(-aOn(-^)_ _r(y)r_(l-a)_
n(y-a'-|3')n(-y) rO+y)r(y+8"-i)'

then

Substituting, we have

By using the properties of the r functions, the coefficient of 7"4 can be proved
equal to

35—2

548 LIMITING CASE OF POLYGON [269.

and the coefficient of F2 can be proved* equal to

e"ia r (V) T (8} 0"*" sin yn , T (y) T (8)

—

Moreover

«', i-?, a-y. /•},

1-,.}
F2 .P 1-a + S

and therefore an equation to determine p. is

c r(/3)r(y+a)

^
l-o, y + ftl-/i} 6 T (8) T (y + j8) '

Ex. A regular polygon of n sides, in the ?0-plane, has its centre at the origin and one
angular point on the axis of real quantities at a distance unity from the origin. Shew that
its interior is conformally represented on the interior of a circle, of radius unity and centre
the origin, in the 2-plane by means of the relation

n -- /•• --

w I (l-xn) ndx=\ (l-2n) ndz. (Schwarz.)

Jo Jo

270. It is natural to consider the form which the relation assumes when
we pass from the convex polygon to a convex curve, by making the number
of sides of the polygon increase without limit. The external angle between
two consecutive tangents being denoted by d^r, and the internal angle of the
polygon at the point of intersection of the tangents being £TT, we have

7T — 7T =

so that £ — 1 — ~~ ~~ •

Let x be the point on the axis of real quantities, which corresponds to this
angular point of the polygon ; then the limiting form of the relation

d /, dw\ v a — 1

-y- I log -y- = 2, —

dz \ dz] z — a
d f, dw\ 1 f dty

where x is the point on the real axis in the ^-plane corresponding to the
point on the w-curve at which the tangent makes an angle -ty with some
fixed line, and the integral extends round the curve, which is supposed to be
simple (that is, without singular points) and everywhere convex.

The disadvantage of the form is that x is not known as a function of ty,
and its chief use is to construct curves such that the contour is conformally
represented, according to any assigned law, along the axis of real quantities

* In reducing the coefficients to these forms, limiting cases (such as /3 + y = l) of the quadri
lateral are excluded.

270] AS A CONVEX CURVE 549

in the ^-plane. The utility of the form is thus limited : the relation is not
available for the construction of a function by which a given convex area in
the w-plane can be conformally represented on the half of the ^-plane*.

Ex. Let # = tan£^: then taking the integral from - TT to +TT, we have

2 fin1 d(f)
TT J - jn- z — tan
The integral on the right-hand side is

J o z — tan <£ ] \itZ-\- tan 0

o z2 - tan2 <j>

f«

=2*1

7o

f-ji-^j-i,,,,

jo U+y y +(2^)2J *

and therefore ^ Aog ^ = _ A

c?2 V 5 dz) z-i'

which, on further integration, leads to the ordinary expression for a circle on a half-
plane.

271. In regard to the conformal representation on the half of the /-plane
of figures in the w-plane bounded by circular arcs, we proceed f in a manner
similar to that adopted for the conformal representation of rectilinear polygons.

It is manifest that, if u =f(z) determine a conformal representation on
the ^-plane of a w-polygon bounded by circular arcs and having assigned
angles, then

Au + B

where A, B, G, D may be taken subject to the condition AD -BC=l, will
represent on the half s-plane another such polygon with the same assigned

* See Christoffel, Gott. Nadir., (1870), pp. 283—298.

t For the succeeding investigations the following authorities may be consulted :—

Schwarz, Ges. Werke, t. ii, pp. 78—80, 221—259.

Cayley, Camb. Phil. Trans., vol. xiii, (1879), pp. 5—35.

Klein, Vorlesungen iiber das Ikosaeder, Section I., and particularly pp. 77, 78.

Darboux, Theorie generate des surfaces, t. i, pp. 180—192.

Klein-Fricke, Theorie der clliptischen Modulfnnctionen, t. i, pp. 93—114.

Goursat, I.e., p. 546, note.

550 SCHWARZIAN DERIVATIVE [271.

angles: for the homographic transformation, preserving angles unchanged,
changes circles into circles or occasionally into straight lines. Hence, as
in § 264, when the transforming function is being obtained, it is to be expected
that it will be such as to admit of this apparent generality : and therefore,

since

{w, z\ = \u, z},

where {w, z\ is the Schwarzian derivative, it follows that, in obtaining the
conformal representation of a figure bounded by circular arcs, the function to
be constructed is

0 , , w" ,/w
8 = {w, z \- —f - 1 —
w i \w

We proceed as in the case of the rectilinear polygon and find the form of
the appropriate function in the vicinity of points of various
kinds. But one immediate simplification is possible, which
enables us to use some of the earlier results.

Let C be an angular point, CA and CB two circular
arcs, one of which may be a straight line : if both were
straight lines, the modification would be unnecessary. In-
vert the figure with regard to the other point of intersection
of CA and CB : the two circles invert into straight lines cutting at the same
angle ///TT. Take the reflexion of the inverted figure in the axis of imaginary
quantities : and make any displacement parallel to the axis of real quantities:
if W be the new variable, the relation between w and W is of the form

aW+b

where ad — be = 1 ; and therefore

_
cW+d~W'

Consider the function for the TF-plane. Let T be the point corresponding
to (7,vari angular point of the polygon, having z — c as its homologue on the
axis of x, account being taken of the possibility of having c — <x> ; let /3 be any
point on either of the straight lines corresponding to a point on the contour
of the polygon not an angular point, having z = b as its homologue on the
axis of x. If a contour point not an angular point have 0 = 00 as its
homologue on the axis, denote it by /3'.

Then for the vicinity of /3, we have (as in § 267) a relation of the form
TT - £ = ei(7r+0> (z-b}P(z-b}:

dW
then log -y- = const. + log Px (z — b),

so that {W, z}=P2(z-b),

where P2 is an integral function of z — b, converging for sufficiently small
values of \z — b\.

271.] FOR REPRESENTATION ON A CIRCLE 551

For the vicinity of ft', we have similarly

then

z \z
dW . 1

and therefore { W, z} = — -I

2 ~v / 1

where Q,, does not vanish for z — oo .

In the vicinity of the angular point F, having a finite point on the axis of
se for its hornologue, we have

W - F = e^^ (z - cY R(z- c),
and, proceeding as before, we find that

,

^ — c- ^ — c

where C0 depends on the coefficients in the series R (z — c).

But if the angular point F have the point at infinity on the axis of x for
its homologue, we have

then, proceeding as before, we find that

where Tn (-} does not vanish when z = oo .

~ \zJ

Lastly, for a point W in the interior having its homologue at z = z', we

have

W- W' = (z-z'}

and then {W, z} = S,(z-z').

Hence { W, z}, considered as a function of z, has the following properties : —

(i) It is an analytical function of z, real for all real values of the
argument z ; and if x = oo do not correspond to an angular
point of the polygon, then for very large values of z

where Q., is finite when z = x .

552 CURVILINEAR POLYGON [271.

(ii) It has a finite number of accidental singularities, all of them
isolated points on the axis of x : and at all other points on one
side of the plane it is uniform finite and continuous, having
(except at the accidental singularities) real continuous values
for real continuous values of its argument. Its form near the
singularities, and its form for infinitely large values of z, if
z = oo be the homologue of an angular point, are given above.

Hence {W, z\ can be continued across the axis of x, conjugate values of
{W, 2} corresponding to conjugate values of z: and thus its properties make
it an algebraical rational meromorphic function of z.

Two cases have to be considered.

First, let the angular points of the polygon have their homologues at
finite distances from the ^-origin, say, at a, b, ...,l: and let CLTT, /3?r, ..., XTT be
the internal angles of the polygon at the vertices. Then

z — a z — a

has no infinity in the plane ; it is a uniform analytical function of z, and
must therefore be a constant, which, by the value at z = oo , is seen to be
zero. Hence

the summation being for the homologues of all the angular points of the
polygon. But when z is very large, we have, in this case

[W,z}=±(
so that, expanding 2J (z) in powers of- and comparing with the latter form,

Z

we have, on equating coefficients of z~l, z~"-, z~3,

0 = 2A0a + i 2 (1 - a2),

relations among the constants of the problem.

Secondly, let one angular point, say a, of the polygon have its homologue
on the axis of x at infinity, and let O.TT be the internal angle at a : and let the
homologues of the others be b, ...,k, I, the internal angles of the polygon
being fiir, . . . , KTT, XTT. Then the function

[W,z}-^^-^}-^~
z-b 2 (z - b)-

271.] REPRESENTED ON A CIRCLE 553

has no infinity in the plane : it is a uniform analytical function of z, and
must therefore be a constant, say M ; thus

But, when z is very large, we have

because x — oo is the homologue of the vertex a of the polygon, the angle
there being our: and 2*(-j does not vanish when z=oo. Hence, expanding

\2 )

in powers of - and comparing coefficients, we have

M = 0,

so that . {F,,HS + ^2

where the summation is for the homologues of all the angular points other
than a, and the constants are subject to the two conditions

The form of the function { W, z} is thus obtained for the two cases, the
latter being somewhat more simple than the former : and the exact expansion
of W in the vicinity of a singular point can be obtained with coefficients
expressed in terms of the constants.

272. In either case the equation which determines W is of the third
order : but the determination can be simplified by using a well-known
property of linear differential equations*. If yl and y., be two solutions
of the equation

the quotient of which is equal to the quotient of two solutions of

where I=Q — , -- P2, being the invariant of the equation for linear trans

formation of the dependent variable, and where Yjy = e^pdx, then the equation
satisfied by s, = 3/1/2/2, is

(«,*}= 27.

* See my Treatise on Differential Equations, pp. 89 — 93.

554 CRESCENT [272.

Hence for the present case, if we can determine two independent solutions
J^i and Z.> of the equation

for the first case, or two independent solutions of the equation

for the second case, then

is the general solution of the equation

or

and therefore is the function by which the curvilinear w-polygon is conform-
ally represented on the ^-half-plane.

273. As a first example, consider the w-area between two circular arcs
which cut at an angle XTT. The ^-origin can be conveniently taken as the
homologue of one of the angular points : aud the ^-point at infinity along the
axis of x as the homoloue of the other. Then we have

provided 4=0, 4.0 = i(l-X2)-£(l- X2),

both of which conditions are satisfied by A = 0 ; and so

f

The linear differential equation is

dz° z-

cr> tViot 7. — «*(1 + A) 5f — ~t(l-A) .

nO I mil Z/i — /3- , Z/2 — •* 5

and therefore the general solution for W is

W = ^1±J.

cz* + a

The (three) arbitrary constants can be determined by making z = 0 and
z = oo correspond to the angular points of the crescent, and the direction of
the line z = z0 (which is the axis of x) correspond to one of the circles, the
other of the circles being then determinate.

If the w-circles intersect in — i (the homologue of the ^-origin) and + i

273.] CURVILINEAR TRIANGLE 555

(the homologue of x = oo ), and if the centre of one of the circles be at
the point (cot a, 0), then the relation is

^A ^^ />^ — O.1

. 6 \jU

W=l -^ 3^ ,

where c is an arbitrary constant, equivalent to the possible constant magnifi
cation of the 2-planc without affecting the conformal representation : it can
be determined by fixing homologous points on the contour of the crescent.

More generally, if the w-circles intersect in wr and w», respectively homo
logous to z = 0 and z = <x> , then

w — w.,
is the form of the relation.

Evidently a segment of a circle is a special case.

274. Next, consider a triangle in the w-plane formed by three circular
arcs and let the internal angles be XTT, /ATT, VTT. The homo- ,

logue of one of the angular points, say of that at /XTT, can be .s^paKT

taken at z = oo ; of one, say of that at XTT, at the ^-origin ; and of
the other, say of that at VTT, at a point z = 1 : all on the axis

of x. Then we have

>> \^_ VYI

= ? + a +ii^ + ilz±l ^TV

where the constants B and G are subject to the relations

B . 0 + C. 1 = i (1 - /x2) - -| (1 - X2) - i (1 - v-},

so that — B = C = \ (X2 — jj," + v" — 1),

and therefore

j ] 1 — X2 , 1 — v2 , X2 — /i2 + v2 — 1

\ ) 4- o I 4- / i \o i 4- / ~t \ *

z- (z — I)2 z (z — 1)

But / (z) is the invariant of the differential equation of the hypergeometric

series* ;•

~~j 9 ' 7i \ TT ~7~\ \^ """ '

provided X2 = (1 — j)2, \£ = (a — /3)2, v2 = (7 — a — /3)2 ;

so that, if Z1 and Z.2 be two particular solutions of this equation, the function
which gives the conformal representation of the w-triangle on the ^-half-
plane is

AZ, + BZ,

W= 7777-

Differential Equations, p. 188.

556

CONFORMAL REPRESENTATION

[274.

The transforming function thus depends upon the solution
equation of the hypergeometric series, and for general
which are > 0 < 1 we shall obtain merely general values
the transforming function will be obtained as a quotient
solutions of the equation of the series. Now according to

of the differential
values of \, p,, v
of a, j8, 7; hence
of two particular
the magnitude of

o

, these solutions, which are in the form of infinite series, change : and thus
we have w equal to an analytical function of z, which has different branches
in different parts of the plane.

The distribution of the values z = 0, 1 , GO as the homologues of the three
angular points was an arbitrary selection of one among six possible arrange
ments, which change into one another by the following scheme : —

1

1

2

2-1

2

1-2

-

2

1-2

2-1

2

0

1

00

1

0

00

1

0

1

00

00

0

oo

00

0

0

1

1

The quantities in the first row are the homographic substitutions, conserving
the positive half-plane and interchanging the arrangements.

These substitutions are the functions of z subsidiary to the derivation of
Kummer's set of 24 particular solutions of the equation of the hypergeometric

series.

Ex. Take the case when two of the angles of the triangle are right, say v = %, /LI = £,
. = -. Then, when n is finite*, a transforming relation is

..l-(l-«)*.

and, when n is infinite, a transforming relation is

, --
w = \og

5

obtainable either as a limiting form of the above, or by means of the solutions F (a, /3, y, 2)
and F(a, ft, a + ft — y + 1, 1-2) of the differential equation of the hypergeometric series.
In the respective cases the general relations, establishing the conformal representation, are

and

\cw -f dj

aw + (

cw + d
Differential Equations, p. 208.

274]

OF CURVILINEAR TRIANGLE

557

O'

Fig. 98.

The three circles, arcs of which form the triangle, divide the whole of the
w;-plane into eight triangles which can be arranged
in four pairs, each pair having angles of the same
magnitude. Thus

-D, D' have angles ATT, /ATT, VTT,

A, A' ATT, (1 — /A) TT, (1 — i/) TT,

B, B' (1 — A) 7T, /L47T, (1 — v) 7T,

and G, C' (1 - A)TT, (1 — /K)TT, VTT;

and when any one of the triangles is given, it

determines the remaining seven. It is convenient

then to choose that one which has the sum of its angles the least, say the

triangle of reference : let it be D. Unless A, /j,, v, each of which is > 0 < 1, be

each — ^, then A + p* + v < f .

We have already, in part, considered the case in which A + p + v = 1.
For, when this equation holds, inversion with the other point having ATT for
its angle as centre of inversion, changes* D into a triangle bounded by
straight lines arid having XTT, /JLTT, vir as its angles; and therefore, in that
case, the problem is merely a special instance of the representation of a
w-rectilinear polygon on the ^-half-plane.

But there is a very important difference between the cases for which
A 4- n + v < 1 and those for which A + fj, + v>I: in the former, the ortho
gonal circle (having its centre at the radical centre of the three circles) is real,
and in the latter it is imaginary. The cases must be treated separately.

275. First, we take A + p + v < 1. Then of the two triangles, which
have the same angles, one lies entirely within the orthogonal circle and the
other entirely without it ; and each is the inverse of the other with regard to
the orthogonal circle f. Let inversion with regard to the angular point ATT in
A take place : then the new triangle is bounded by two straight lines cutting
at an angle ATT and by a circular arc cutting them at
angles /U.TT and vir respectively, the convex side of the
arc being turned towards the straight angle. The
new orthogonal circle is the inverse of the old and its
centre is A, the angular point at ATT ; its radius is the
tangent from A to the arc CB, and therefore it com
pletely includes the triangle ABC.

The homologue of A is, as before, taken to be the ^-origin 0, that of G to
be the point z = 1, say c, and that of B to be z = oo on the axis of x, say b for
-f oo and b' for — oo .

* The figure in the text does not apply to this case, because, as may easily be proved, the three
circles must meet in a point.

t For the general properties of such systems of circles, see Lachlan, Quart. Journ. Math., vol.
xxi, (1886), pp. 1—59.

Fig. 99.

558 FUNCTIONAL RELATION [275.

Suppose that we have a representation of the triangle on the positive
half-plane of z. The function {w, z\ can be continued across the axis of x
into a negative half-plane, if the passage be over a part of that axis, where
the function is real and continuous, that is, if the passage be over Oc, or over
cb, or over b'O ; arid therefore w is defined for the whole plane by {w, z} = '2I(z)>
its branch-points being 0, c, b. Any branch on the other side, say w1} will
give, on the negative half-plane, a representation of a triangle having the
same angles, bounded by circular arcs orthogonal to the same circle, and
having 0, c, b for the homologues of its angular points. Thus if the con
tinuation be over cb, the new w-triangle has CB common with the old, and
the angular point A' lies beyond OB from A.

To obtain the new triangle A'GB geometrically, it is sufficient to invert
the triangle ACB, with regard to the centre of the circular arc CB. This
inversion leaves CB unaltered ; it gives a circular arc CA' instead of CA
and a circular arc BA' instead of BA : the angles of A'CB are the same as
those of ACB. Since the orthogonal circle of ACB cuts CB at right angles
and CB is inverted into itself, the orthogonal circle is inverted into itself;
therefore the triangle A'CB has the same orthogonal circle as the triangle
ACB.

The branch w1 , by passing back across the axis round a branch-point into
the positive half-plane, leads to a new branch w2, which gives in that half-plane
a representation of a triangle, again having the angles XTT, /ATT, VTT and having
0, c, b for the homologues of its angular points. Thus if the passage be
over Oc, the new ?/;-triangle has A'C common with A'CB and the angular
point B" lies on the side of CA' remote from B: but if the passage be
over cb, then we merely revert to the original triangle CAB. The new
triangle has, as before, the same orthogonal circle as A'CB.

Proceeding in this way by alternate passages from one side of the
axis of x to the other, we obtain each time a new w-triangle, having one side
common with the preceding triangle and obtained by inversion with respect
to the centre of that common side : and for each triangle we obtain a new
branch of the function w, the branch-points being 0,1, oo . If, by means of
sections such as Hermitc's (§ 103), we exclude all the axis of # except the part
between two branch-points, the function is uniform over the whole plane thus
bounded.

All these triangles lie within the orthogonal circle, and they gradually
approach its circumference : but as the centres of inversion always turn that
circle into itself, while the sides of the triangle are orthogonal to it, they do
not actually reach the circumference. The orthogonal circle forms a natural
limit (§ 81) to the part of the w-plane thus obtained.

Ex. Shew that all the inversions, necessary to obtain the complete system of triangles,
can be obtained by combinations of inversions in the three circles of the original triangle.

(Burnside.)

275.] FOR CURVILINEAR TRIANGLE 559

Each of the triangles, thus formed in successive alternation, gives a
w-region conformally represented on one half or on the other of the ^-plane.
If, then, the original triangle be combined with the first triangle that is
conformally represented on the negative half-plane, every other similar
combination may be regarded as a symmetrical repetition of that initial
combination: each of them can be couformally represented upon the whole
of the ^-plane, with appropriate barriers along the axis of x.

The number of the triangles is infinite, and with each of them a branch
of the function w is associated : hence the integral relation between w
and z which is equivalent to the differential relation {w, z\ = 21 (z), when
X + fj, + v < 1, is transcendental in w.

In the construction of the successive triangles, the successive sides passing
through any point, such as G, make the same angle each with its predecessor:
and therefore the repetition of the operation will give rise to a number of
triangles at C each having the same angle XTT.

If A, be incommensurable, then no finite number of operations will lead to
the initial triangle : each operation gives a new position for the homologous
side and ultimately the w-plane in this vicinity is covered an infinite number
of times, that is, we can regard the w-surface as made up of an infinite
number of connected sheets.

If X be commensurable, let it be equal to l/l', where I and I' are integers,
prime to each other. When I is odd, 21' triangles will fill up the w-space
immediately round (7, and the (21' + l)th triangle is the same as the first : but
the space has been covered I times since 2/'X?r = 21-Tr, that is, in the vicinity of
C we can regard the w-surface as made up of I connected sheets. When I is
even (and therefore I' odd), I' triangles will fill up the space round G com
pletely, but the (I' + l)th triangle is not the same as the first : it is necessary
to fill up the space round C again, and the (21' + l)th triangle is the same as
at first ; the space has then been covered I times, so that again the w-surface
can be regarded as made up of I connected sheets. The simplest case is
evidently that, in which X is the reciprocal of an integer, so that 1=1;
and the w-surface must then be regarded as single-sheeted.

Similar considerations arise according to the values of p and of v.

If then either X, /z, or v be incommensurable, the number of w-sheets is
unlimited, that is, z as a function of w has an infinite number of values, or the
equation between z and w is transcendental in z. Hence, when X + /* + v < 1
and either X or fj, or v is incommensurable, the integral relation between w and
z, which is equivalent to the differential relation {w, z\ = 21 (z), is transcend
ental both in w and in z.

If all the quantities X, p, v be commensurable and have the forms I/I',
mfm', n/n', fractions in their lowest terms, and if N be the least common
multiple of I, m, n, then the number of w-sheets is N, that is, z as a function

560 SPECIAL [275.

of w has N values and therefore the equation between z and w is algebraical
in z, of degree N. Hence, when X + p + v < 1 and X, /j,, v have the forms of
fractions in their lowest terms, and if N be the least common multiple of their
numerators, the integral relation between w and z equivalent to the differential
relation

is an algebraical equation of degree N in z, the coefficients of which are
transcendental functions of w.

The simplest case of all arises when X, /A, v are the reciprocals of integers :
for then N = 1 and z is a uniform transcendental function of w, satisfying
the equation

{«;,*} = 27 (*);

or, making z the dependent and w the independent variable, we have the
result : —

A function z that satisfies the equation

1 111

dsz dz 3 /^fVl
dw3 dw * \dw~j J

dz
dw

z(z-l)

where I, m, n are integers, such that T H h - < 1, is a uniform transcendental

I m n

function ofw.

Restricting ourselves to the last case, merely for simplicity of explanation,
it is easy to see that the whole of the space within the orthogonal circle is
divided up into triangles, with angles XTT, /ATT, VTT bounded by circular arcs
which cut that circle orthogonally : and, by the inversion which connects the
space external to the circle with the internal space, the whole of the outside
space is similarly divided. Moreover, it has been seen that every triangle

can be obtained from any one by some substitution of the form wr = — ~ :

crw + dr

therefore the division of the interior of the circle into triangles is that
which is considered, in the next chapter, for the more general case of division
into polygons, the orthogonal circle of the present case being then the
' fundamental ' circle. The uniform transcendental function of w is therefore
automorphic : the infinite group of substitutions is that which serves to
transform a single triangle into the infinite number of triangles within the
circle *S

One or two special cases need merely be mentioned.

If any one of the three quantities X, /z, v be zero and if X + p + v is
not equal to unity, the triangle can be included under the general case
just treated. For let X = 0, and suppose that /A + v is not greater than unity :

* The figure for the example v = %, fJ-—\, A = l is given by Schwarz, Ges. Werke, t. ii, p. 240;
and the figure for the example v = \, M = i. X=f is given in Klein-Fricke (p. 370) ; both of course
satisfying the conditions X + /u + v < 1.

275.] CASES 561

if //, + v were greater than unity, the triangle would be a particular instance
of the class about to be discussed. The division of the area within the
(real) orthogonal circle is of the same general character as before : a
particular illustration is provided by the division appropriate to the
elliptic modular-functions, for which [*> = $, v = ^ (§ 284). When two
triangles, one of which is obtained from the other by continuation in the
£-plane across the axis of real variables, are combined, they give a w-space
(corresponding to the whole of the 2-plane) for which X = 0, At' = i, i> = i.

\JLO JL / ' ' O' o

Since the orthogonal circle is real, it forms a natural limit to these spaces ;
when it is transformed into the axis of real variables in the w-plane by
a homographic substitution, the positive half of the w-plane is divided as
in figure 108 (p. 590).

The extreme case of the present class, for which X + /m + v is less than
unity, is given by X = 0, ^ = 0, v = 0 : the triangle is then the area between
three circles which touch one another. Reverting to the differential equa
tion of the hypergeometric series, we have 7 = 1, a = /3 = ^; the equation is

ff+4^" -^_,=0,

dz* Z(L — z) dz z(i—z)

which is the differential equation of the Jacobian quarter-periods in elliptic
functions with modulus equal to z^. If

K = I ' (I - z sin2 </>)-* d(f>, K' = f {1 - (1 - z) sin2 <£}-* d<j>,
Jo Jo

then K/

w =

aK + bK'
or, more generally, w = — ^ — -™, ,

C-t\. "r u/J\.

a relation between w arid z which gives the con formal representation of the
w-triangle upon the ^-half-plane.

276. We now pass to the consideration of the case in which the triangle
with angles XTT, /ATT, VTT has no real orthogonal circle : the other associated
triangles have therefore not a real orthogonal circle. In this case, the sum of
the angles of the triangle is greater than TT, so that we have

X + fj, + v > 1 from the pair D and D',

— \ + lj, + v<l from the pair A and A',

\ — fji + v < 1 from the pair B and B',

\ + fj, — v < 1 from the pair C and C",

as the conditions which attach to the quantities X, /LI, v. As before, we invert
F. 36

562 STEREOGRAPHIG PROJECTION [276.

with respect to the angular point \TT in A : then the new triangle D is
bounded by two straight lines and a circle, the
intersection of the lines being in the interior of the
circle, because the orthogonal circle is imaginary.
Let d be distance of L from the centre of the
circle, 6 the angle OLN, r the radius of the circle :
then
d sin 6 = — r cos VTT, d sin (\TT — 6) = — r cos /ATT,

which determine d and 6. Let R2 = r* — d?, so that

iR is the radius of the (imaginary) orthogonal circle. Fig. 100.

With L as centre and- radius equal to R describe a sphere : let P be
the extremity of the radius through L perpendicular to the plane. Then P ,
can be taken as the centre for projecting the plane on the sphere stereo- j
graphically*; so that, if Q be a point on the plane, Q' its projection
on the sphere, PQ . PQ' = 2E2. The projection of LN is a great circle
through P, the projection of LM is another great circle through P in
clined at XTT to the former: and since PO is equal to the radius of the
plane circle, so that its diameter subtends a right angle at P, the stereo- 1
graphic projection of that plane circle is a great circle on the sphere, I
making angles VTT and /-ITT with the former great circles. There is thus,
on the sphere, a triangle bounded by arcs of great circles, that is, a
spherical triangle in the ordinary sense, whose angles are XTT, yu/7r, VTT : and
this spherical triangle is conformally represented on the ^-half-plane, its-
angular points L, N, M finding their homologues in z = 0, 1, oo respectively.

Just as in the former case, the successive passages, backwards
and forwards across the ^-axis, give in the w-plane new triangles with
angles XTT, /ATT, VTT, all with the same imaginary orthogonal circle of
radius iR and centre L : each of these, when stereographically projected *
on the sphere with P as the centre, becomes a spherical triangle of angles
XTT, /ATT, VTT bounded by arcs of great circles, every triangle having one side
common with its predecessor : and the triangles are equal in area.

Moreover, the triangles thus obtained correspond alternately to the
positive half and the negative half of the 0-plane : and it is convenient to
consider two such contiguous triangles, connected with the variable w,
as a single combination for the purposes of division of the spherical
surface, each combination corresponding to the whole of the 2-plane.

The repetition of the analytical process leads to the distribution of the
surface of the sphere into such triangles : and the nature of the analytical
relation between w and z depends on the nature of this distribution.

If X, fju, or v be incommensurable, then the number of triangles is
* Lachlan, (I.e., p. 557, note), p. 43.

276.] DIVISION OF SPHERICAL SURFACE 563

infinite, so that the relation is transcendentaJ_jn_jz£LL_and the surface of
the sphere is covered an infinite number of times; that is, corresponding
to z there is an infinite number of sheets, so that the relation is trans
cendental in z. Thus, when \ + JJL + v is greater than 1 and any one of
the three quantities X, /A, v is incommensurable, the integral relation
between w and z, which is equivalent to

{«;,*) = 27 (*),
is transcendental both in w and in z.

If the quantities X, //,, v be commensurable, the simplest possible
cases arise in connection with the division of the surface by the central
planes associated with the inscribed regular solids. These planes give the
divisions into triangles, which are equiangular with one another.

First, suppose that the spherical surface is divided completely and
covered only once by the two sets of triangles, corresponding to the upper
half and the lower half of the 5-plane respectively. One of the sets, say
N in number, will occupy one half of the surface in the aggregate : and
similarly for the other set, also N in number. Hence

R2(\ + jj, + v — 1 ) TT = the area of a triangle

= jj. (area of a hemisphere),

2
so that \+ /ju + v— 1 = -*? •

Then, in passing round an angular point, say XTT, the triangles will
alternately correspond to the upper and the lower halves : hence, of the
whole angle 2?r, one half will belong to one set of triangles and the
other half to the other set. Hence TT -f- XTT is an integer, that is, X is the

reciprocal of an integer, say j j Similarly for p,, which must be of the form

I .

— ; and for v, which must be of the form - ; where m and n are integers.
m n

Thus

i+1+1_l=l.

I m n N'

The only possible solutions of this equation are

(I.)* X = £, /A = £, n = any integer, N=2n;

(II.) X = i, p = l, v = ^ , JV=12;

(IV.) X = i, /. = !, v = ^ , JV=24;

(VI.) X = i, /.= ', v = t , ^=60.

277. In each of these cases there is a finite number of triangles : with
each triangle a branch of w is associated, so that there is only a finite number

* The reason for the adoption of these numbers to distinguish the cases will appear later, in
§279.

36—2

564

SPHERICAL SURFACE

[277.

Fig. 101.

of branches of w : the sphere is covered only once, and therefore there is only
a single ^-sheet. Hence the integral relation between w and z is of the first
degree in z : and it is algebraical in w, of degrees 2?i, 12, 24, 60 respectively.

The regular solids, with which these sets of triangles are respectively
associated, are easily discerned.

I. We have X, fju, v = ^, -|, -. The solid is a double pyramid, having

its summits at the two poles of the sphere : the
common base is an equatorial polygon of 2w sides :
the sides of the various triangles, in the division of
the sphere, are made by the half-meridians of longi
tude, through the angular points of the polygon
from the respective poles to the equator, and by arcs
of the equator subtended by the sides of the polygon.

II. We have X, p, v = \, £, £. The solid is the
tetrahedron ; and the division of the surface of the
sphere, by the planes of symmetry of the solid, into

24 triangles, 12 of each set, is indicated, in fig. 102, on the (visible) half of
the sphere, the other (invisible) half of the sphere being the reflexion, through
the plane of the paper, of the visible half.

The angular summits of the tetrahedron are T, the middle points of its
edges are S, the centres of its faces are F : all
projected on the surface of the sphere from
the centre. If desired, the summits of the
tetrahedron may be taken at F: the centres
of the faces are then T.

Each of the angles at T is ^TT : each of the
angles at F is ^TT: each of the angles at S
is |TT.

The shaded triangles (only six of which
are visible, being half of the aggregate) corre
spond to one half of the ^-plane; and the un
shaded triangles correspond to the other half
of the £-plane.

iTv. We have X, p, v = ^, ^, {. The solid is the cube or the octahedron.
These two solids can be placed so as to have the same planes of symmetry, by
making the centres of the eight faces of the octahedron to be the summits of
the cube. In the figure (fig. 103), the points 0 are the summits of the
octahedron : the points C are the summits of the cube and the centres of the
faces of the octahedron : and the points S are the middle points of the edges :
all projected from the centre of the sphere.

277.]

AND REGULAR SOLIDS

565

The shaded triangles (the visible twelve being one half of the aggregate)
correspond to one half of the 5-plane ; the unshaded triangles correspond to
the other half of the ^-plane.

Each of the angles at 0 is ITT : each of the angles at G is ^TT : each of the
angles at S is |TT ; and it may be noted that the triangles COO are the
triangles in the tetrahedral division of the spherical surface, the point 0 in
the present triangle COG being the point S in a triangle STF and the
two points G being the points F and T in the former figure (fig. 102).

VI We have \, p, v =

, 1.

D,5

566 CONSTRUCTION [277.

The solid is the icosahedron or the dodecahedron. These two solids can
be placed so as to have the same planes of symmetry, by making the centres
of the twenty faces of the icosahedron the vertices of the dodecahedron. In
the figure (fig. 104) the vertices of the icosahedron are the points 7: those
of the dodecahedron are the points D : and the middle points of the edges are
the points 8. The shaded triangles (the visible thirty, six in each lune
through a vertex of the icosahedron, being one half of their aggregate)
correspond to one half of the ^-plane : the unshaded triangles, equal in
number and similarly distributed, correspond to the other half of the ^-plane.
The angles at the vertices 7 of the icosahedron are £TT ; those at the vertices
7) of the dodecahedron are |TT ; and those at the middle points 8 of the edges
(the same for both solids) are \ir.

278. Having obtained the division of the surface, we now proceed to
determine the functions, which establish the conformal representation.

In all these cases, z is a uniform algebraical function of w : therefore
when we know the zeros and the infinities of z as a function of w, each in its
proper degree, we have the function determined save as to a constant factor.
This factor can be determined from the value of w when z=\.

The variable^ belongs to the stereographic projection of the point of the
spherical surface on the equatorial plane, the south pole being the pole of
projection. If X, Y, Z be the coordinates of the point on the spherical
surface, the radius being unity, then

X + iY
'TTT-

For a point in longitude I and latitude ^TT — 8, we have X = cos I sin 8,
Y= sin I sin 8, Z= cos 8 : so that, if preferable, another form for w is

w = ea tan \ 8.

In our preceding investigation, the angle at XTT was made to correspond
with z = 0, that at vir with z = l, that at /ATT with z — oo .

Case I. We take X = -,//,= 1 v=\.

n '

For the angular points pir we have 8 = \TT ; I = 0, — , -—,..., each point

belonging to two triangles of the same set, that is, triangles represented on
the same half of the plane : thus the various w-points in the plane are

for r = 0, 1, . . ., n — 1, each occurring twice. Hence z = oo , when the function

n-l ?7nr

II (w-en )2

r=0

vanishes, that is. z= x , when (wn — I)2 vanishes.

278.] OF TRANSFORMING RELATIONS 567

For the angular points VTT, we have S = ^7r; 1 = -, — , - — ..... each

n n n

point belonging to two triangles of the same set : thus the various w-points in
the plane are

e5
for r = 0, 1, . . ., n — I, each occurring twice. Hence z = 1, when the function

r=0

vanishes, that is, z = I, when (wn + I)2 vanishes.

Now z is a uniform function of w : hence we can take

\-z=K(n"

(wn - I)2 '

where K is a constant, easily seen to be unity : because, when w = 0 (corre
sponding to the common vertex A/TT at the North pole) and when w = oo (corre
sponding to the common vertex \TT at the South pole), z vanishes, as required.
The relation is often expressed in the equivalent form

z : z - 1 : 1 = - 4<wn : - (wn + I)2 : (wn - I)2,
which gives the conformation on the half ^-plane of a w-triangle bounded by

TT

circular arcs, the angles being -, ITT, £TT. The simplest case is that in

TT

-

IV

which the triangle is a sector of a circle with an angle — at the centre.

?z

The preceding relation is a solution of the equation

If we choose \ = £, /& = £, i/=-; so that ^ = 0, when (wn + Vf vanishes,

z=cc, when (w?l-l)2 vanishes, and ^ = 1, when wn vanishes, the relation
establishing the conformal representation will be

z\z-\ : 1 = (wn + I)2 : 4,wn : (wn - I)2 :
this relation is a solution of the equation

Case II. We take X = \ • so that 2 = 0 must give the points S, each of
them twice, since there are two triangles of the same set at S : n = ± (and
these are taken at T), so that z = oo must give the points T, each of them

568 TETRAHEDRAL FUNCTION [278.

thrice : and v = | (and these are taken at F}, so that z = 1 must give the
points F, each of them thrice.

Taking the plane of the paper as the meridian from which longitudes are
measured, the coordinates of the four w~points in the plane, corresponding to
T by stereographic projection, are

V2 _ V2 . V2 _ . V2

yo /o /o ' /O

A/ O \/ O \/ O W *r

i~' T' i ' T'

\ \ — __ \ _j_ j i

\/3 \/3 \/3 \/3

say Wj, w2, w3, w4. Then ^=00 gives each of these points thrice: that is,
z = oo , when {(w — w^). ..(w — w4)}3 vanishes, or z = oo , when

(w* — 2 w2 v8 — 1 )3
vanishes.

The coordinates of the four points corresponding to F, are

.IS) /f) tf) If)

V ^ Y^ • v** • V •*

T' T' T' i"'

1-1- 14- 1- 1-

^ /Q "^ /Q /Q /Q

Y«J Y*J V ** V *•

Hence ^=1, when
vanishes.

(2r+l) —

The coordinates of the six points corresponding to 8 are 0, e" 4 (for
r = 0, 1, 2, 3) and oo : hence z=Q, when

vanishes.

Moreover, z is a uniform function of w : and therefore

the constant multiplier on the right-hand side being determined as unity by
the relation between the points S and the value z = 0.

The relation is often expressed in the equivalent form
z : 2-1:1 = 1'2 \/3w- (w4 + I)2 : (w4 + 2w2 V3 - I)3 : - (w4 - 2w2 \/3 - I)3 ;
it gives the conformation on the .z-half-plane of a triangle in the w-plane,
bounded by circular arcs, the angles of the triangle
being ITT, ITT, |TT.

The simplest case is that of a portion cut out ,,-'''

of a sector of a circle of central angle 30°, by the ,,'''

arc and two lines at right angles to one another ^:-

symmetrical with respect to the arc. Fi8- 105-

278.] OCTAHEDRAL FUNCTION 569

It has been assumed that the plane of the paper is the meridian.
Another convenient meridian to take is one which passes through a point
8 on the equator : in that case, the preceding analysis applies if a rotation
through an angle ^TT be made. The effect of this rotation is to give the new
variable W for any point in the form

M

W = we*,

so that w2 = — iW 2. The relation then takes the form
I'.z-l : 1

= 12V^3 F2(TF4-1)2 : (F4 + 2F2v/-3 + l)3:-(F4-2F2v/-"3 + l)3;
but there is no essential difference between the two relations.

The lines by which the ty-plane is divided into triangles, each conformally
represented on one or other half of the z-plane, are determined by z = zn ,
that is, by

(w4 - 2w2 V3 - I)3 (w04-2M;02V3 -I)3

w2(w4+l)2 w0a(w04 + l)*

The figure is the stereographic projection of the division of the sphere, and
it can be obtained as in § 257 (Ex. 13, Ex. 16).

Case IV. We take X = £, so that z = Q must give the eight points C.;
each is given three times, because at C there are three triangles of the same
set : we take v = \, so that z = \ must give the six points 0, each four times :
and p = \, so that z—<x> must give the twelve points 8, each of them twice.

We take the plane of the paper as the meridian. The points 0 are 0, 1,
i, — 1, — i, oo ; each four times. Hence z=\, when the function

[w(w*-l)Y
vanishes.

+ l+i'
The points C are the eight points -~ ~ — : the product of the eight

corresponding factors is

ws + 14w4 + 1 :

and each occurs thrice, so that z = 0, when the function

vanishes.

The points 8 are (i) the four points --=—• - in the plane of the paper,

+ V 2 — 1
giving a corresponding product

w4 - 6w2 + 1 :

_L ,,'

(ii) the four points — = -- in the meridian plane, perpendicular to the

± V 2 - 1

plane of the paper, giving a corresponding product

w4 + 6w2 + 1 :

570

OCTAHEDRAL FUNCTION

[278.

— (2r+l

and (iii) the four points e* , (for r = 0, 1, 2, 3), in the equator, giving a
corresponding product

w4 + l.

Each of these points occurs twice : and therefore z = oo , when the function

{(w4 + 1) O4 - 6w2 + 1) (w4 + 6w2 + I)}2,
that is, when the function

vanishes.
Hence

~ J(V2-33w8-33w4 + I)2 '
the constant multiplier being determined as unity, by taking account of the
value unity for z : and

\-z = -

The relation can be expressed in the equivalent form

z : z-I :l=(u/i+ 14w4 + I)3 : 108w4 (w4 - I)4 : (w12 - 33w8 - 33w4 + I)2 ;
it gives the conformation on half of the ^-plane of a w-triangle bounded by
circular arcs and having its angles equal to ^TT, JTT, £TT respectively.

The lines, by which the w-plane is divided into the triangles, are given by
z = z0, that is, by

(w8 + 14w4 + I)3 _ O08

Fig. 106.

278.] ICOSAHEDRAL FUNCTION 571

The division is indicated in Fig. 106, being the stereographic projection of the
divided spherical surface of Fig. 103, with respect to the south pole, taken
to be diametrically opposite to the central point 0.

Case VI. We take \= ^, so that z = 0 must give the twenty points D,
each of them thrice; v = ^, so that z = \ must give the twelve points /, each
of them five times; and p. = \, so that 2 = 00 must give the thirty points S,
each of them twice.

Let an edge of the icosahedron subtend an angle 0 at the centre of the
sphere : then its length is 2r sin $0. Also, five edges are the sides of a
pentagon inscribed in a small circle, distant d from a summit : hence the
radius of this circle is r sin 6 and the length of the edge is 2r sin 6 sin ITT, so
that

2 sin $0=2 sin 6 sin ITT,

whence tan £0 = -| (V 5 — 1), cot $0 = |(Vo + 1).

ton

Let a denote e10. Then the value of w corresponding to the north pole /
is 0 ; the values of w for the projections on the equatorial plane of the five
points / nearest the north pole are

tan£0, a2 tan 10, a4tan£0, a6 tan $0, a8tan£0:

the values of w for the projections on the equatorial plane of the five points /
• nearest the south pole are

, a3cot|0, a5cot|0, a7cot£0, a9cot£0:

and for projection of the south pole the value of w is infinity. The product
of the corresponding factors is

4 4

w . U (w - of tan $0) . II (w - tfr+l cot $0) . 1

r=0 r=0

= w (w5 — tan5 $6) (w5 + cot5 |#)

= w(yo + iiw5-i)

after substitution. Each point / occurs five times ; and therefore z=\, when
the function

w5(ww + llw5- I)5
vanishes.

The points D lie by fives on four small circles with the diameter through
the north pole and the south pole for axis. The polar distance of the small
circle nearest the north pole is tan 8 = 3 — V-5, and of the circle next to it is
tan 8' = 3 + /v/5, so that

S/15 + 6V5-1

1S -- 1S,

tan £5 = , taniS =

572

POLYHEDRAL

[278.

The function corresponding to the projections of the five points nearest the

north pole is

wr> + tan5 £8,

and to the projections of the five nearest the south pole is

w5 — cot5 ^8 ;
while, for the projections of the other two sets of five, the products are

WK + tan5 £8'

and w5 — cot5 £S'

respectively. Each occurs thrice. Hence z = 0, when the function
{O5 + tan5 |S) (w5 - cot5 p) (ws + tan5 1§') (w5 - cot5 p')}3,
that is, when (w-° - 228w15 + 494w10 + 228?«5 + I)3,

which is the reduced form of the preceding product, vanishes.

Fig. 107.

278.] FUNCTIONS 573

The points 8 lie by tens on the equator, by fives on four small circles
having the polar axis for their axis. Proceeding in the same way with the
products for their projections, it is found that z — oo , when the function

{w30 + I + 522w5 O20 - 1) - lOOOow10 (w10 + I)}2
vanishes.

H _ (w20 - 228w15 + 494w10 + 228w5 + I)3

Z ~ {w30 + 1 + 522w5 (w20 - 1) - lOOOow10 (w10 + l)j2 '
the constant factor being found to be unity, through the value of 1 — z

which is 1 - z = __ 1728w5(w10-r llw5- I)5

~ {w30 + 1 + 522w5 (w20 - 1) - 10005w10 (w10 + I)}2 '

These relations give the conformal representation on half of the ^-plane of a
w-triangle, bounded by circular arcs and having angles \ir, JTT, ITT.

The lines, by which the w-plane is divided into the triangles, are given
by z = z0> that is, by
(w20 - 228M;15 + 494w10 + 228w5 + I)3 _ (w02(1 - 228w015 + 404w010 + 228w05 + I)3

_

w5 (w10 + 1 1 ws - 1 )5 M>o5(w010 + 11 w05 -I)5

The division is indicated in figure 107, which is the stereographic projection* of
the divided spherical surface of figure 104, with /12 as the pole of projection.

279. The preceding are all the cases, in which simultaneously z is a uni
form function of w, and w is an algebraical function of z : they arise -when
the surface of the sphere has been completely covered once with the two sets
of triangles corresponding to the upper half and the lower half of the ^-plane.

But an inspection of the figures at once shews that they are not the only
cases to be considered, if the surface of the sphere may be covered more than

once.

In the configuration arising through the double-pyramid, the surface of
the sphere will be covered completely and exactly m times, if the angles at
the poles be Zm-ir/n, where m is prime to n. The corresponding relation
between w and z is obtained from the simpler form by changing n into n/m.

In the tetrahedral configuration (fig. 102) the surface of the sphere will be
exactly and completely covered twice by triangles FFT (or by triangles TTF,
it being evident that these give substantially the same division of the surface).
The relation between w and z will then be of the same degree, 12, as before
in w, for the number of different triangles in the two w-sheets is still twelve
of each kind : because there are two w-sheets corresponding to the singlp.
z-plane, that relation will be of the second degree in z. The values of the
angles arc, determined by

(in.) a***?***,

In regard to all the configurations thus obtained as stereographic projections of a spherical
surface, divided by the planes of symmetry of a regular solid, Mobius's " Theorie der Symmetr-
ischen Figuren," (Ges. Werke, t. ii, especially pp. 042—699) maybe consulted with advantage;
and Klein-Fricke, Elliptische Modiilfunctionen, vol. i. pp. 102—106.

574 THE FIFTEEN ALGEBRAICAL [279.

Again, in the octahedral configuration, the surface of the sphere will be
exactly and completely covered twice by triangles OCO. The relation
between w and z will be of degree 24 in w and degree 2 in z : and the values
of the angles are determined by

(V.) X,/*,* = f,i,i.

Similarly, a number of cases are obtainable from the icosahedral configu
ration, in the following forms :

(VII.) X, p, v = f , J, ^ with triangles such as

(VIII.) \n,v = lt, i ...........................

(IX.) x,^ = if,i ...........................

(X.) *,»»-tfei ........................... A/i/,;

(XL) X, p, v = 1 f, f ........................... /,/,/,;

(XII.) x,>,*«f,|,i

(XIII.) X,^ = f,i i

(Xiv.) x, ^, „ = i |, i
(XV.) x,^v=f,|,i

Other cases appear to arise : but they can be included in the foregoing, by
taking that supplemental triangle which has the smallest area. Thus,
apparently, /lA^io would be a suitable triangle, with X, p, v = f , f , l : it is
replaced by /12AoAo> an example of case (X.) above.

These, with the preceding cases numbered* (I.), (II.), (IV.), (VI.), form the
complete set of distinct ways of appropriate division of the surface of the
sphere.

It is not proposed to consider these cases here : full discussion will be
found in the references already given. The nature, however, of the relation,
which is always of the form

f(z} = F(w\

where / and F are rational functions, may be obtained for any particular case
without difficulty. Thus, for (III.), we have

when

2:1-2: 1 =-12\/3w2(w4 + l)2 : (w4 + 2w2 \/3 - I)3 : (w4- 2w2V3 - I)3.
Again, if

* These numbers are the numbers originally assigned by Schwarz, Ges. Werke, t. ii, p. 246,
and used by Cayley, Camb. Phil. Trans., vol. xiii, pp. 14, 15.

279.] TRANSFORMING FUNCTIONS 575

a special case of § 278, I., by taking n = l, then

Hence {w, Z} = (-^ [{w, z} - {Z, z}}

_ 16 (Z + 1)2 rKLHr) ,

so that X = A, v = f , /A = ^. Hence the relation

= - 12 V3 w2 (w4 + I)2 : (w4 + 2w2 \/3 - I)3 : (w4 - 2w2 V3 - I)3

gives the conformation of triangles bounded by circular arcs and having
angles JTT, ^TT, f-rr.

The foregoing are the only cases, for \ + //• + v >l, in which the integral
relation between w and z is algebraical both in w and in z.

In all other cases in which X, /i, v are commensurable, this integral
relation is algebraical in z and transcendental in w.

It is to be noticed, in anticipation of Chapter XXII., that, since every
triangle in any of the divisions of the spherical surface, or of the plane,
can be transformed into another triangle, the functions which occur in
these integral relations are functions characterised by a group of substi
tutions. When the functions are algebraical, the groups are finite, and
the functions are then the polyhedral functions : when the functions are
transcendental, the groups are infinite and the functions are then of the
general automorphic type.

The case in which X + /JL + v = 1 has already been considered: the spherical
representation is no longer effective, for the radius of the sphere becomes
infinite and the triangle is a plane rectilinear triangle. The equation may
still be used in the form

with the condition X + /u- + i> = l. A special solution of the equation is then
given by

dw , ,

s = ^-'<i-/r-i,

leading to the result of § 268, the homologue of the angular point /U/TT being

at z = oo .

280. It is often possible by the preceding methods to obtain a relation
between complex variables that will represent a given curve in one plane on

576 FAMILY OF PLANE ALGEBRAICAL [280.

an assigned curve in the other : there is no indication of the character of the
relation for an arbitrary curve or a family of curves. But in one case, at any
rate, it is possible to give an indication of the limitations on the functional
form of the relation.

Let there be a family of plane algebraical curves, determined as potential
curves by a variable parameter* : and let their equation be

F (x, y, u) = 0,

where u is the variable parameter, which, when it is expressed in terms of x
and y by means of the equation, satisfies the potential-equation

&u d*u _
9#2 dy2

Since u is a potential, it is the real part of a function w of x + iy : and the
lines u = constant are parallel straight lines in the w-plane. It therefore
appears that the functional relation between w and z must represent the
w-plane conformally on the ^-plane, so that the series of parallel lines in the
one plane is represented by a family of algebraical curves in the other : let
the relation, which effects this transformation, be

X (z, w) = 0.

Let the algebraical curve, which corresponds to some particular value of u,
say u = 0, be

r(**0)-/(c,y)-A

which in general is not a straight line. Let a new complex £ be determined
by the equation

this equation is algebraical, and therefore £ can be regarded as a function of
w, say ty (w), between which and z, regarded as a function of w, say </> (w),
there is an algebraical equation.

Now when u = 0, z describes the curve

/(a,y) = 0:
hence at least one branch of the function £, defined by

"

* Such curves are often called isothermal, after Lame. The discussion of the possible
functional relations, that lead to algebraical isothermal curves, is due to Schwarz, Ges. Werke,
t. ii, pp. 260—268 : see also Hans Meyer, "Ueber die von geraden Linien und von Kegelschnitten
gebildeten Schaaren von Isothermen ; so wie liber einige von speciellen Curven dritter Ordnung
gebildete Schaaren von Isothermen," (a Gottingen dissertation, Zurich, Ziircher and Furrer,
1879) ; Cayley, Quart. Journ. Math., vol. xxv, pp. 208—214 ; and the memoir by Von der Miihll,
cited p. 500.

280.] ISOTHERMAL CURVES 577

can be taken as equal to x when u — 0, that is, there is one branch of the
function £ which is purely real when w is purely imaginary.

The curves in the 3-plane are algebraical : when this plane is conformally
represented on the £-plane by the foregoing branch, which is an algebraical
function of z, the new curves in the £-plane are algebraical curves, also
determined as potential curves by the variable parameter u. And the £-curve
corresponding to u = 0 is (the whole or a part of) the axis of real quantities.
In order that the conformal representation may be effected by the functions,
they must allow of continuous variation : hence lines on opposite sides of
u — 0 correspond to lines on opposite sides of the axis of real quantities. The
functional relation between £= £ + irj and w = u + iv is therefore such that

£ + ir] — ^r (u + iv),

£ — irj = ty (— u + iv).

The equation of the ^-curves, which are obtained from varying values of
u, is algebraical : and therefore, when we substitute in it for £ and 77 their
values in terms of ^ (u + iv) and -v/r (— u + iv), we obtain an algebraical
equation between ^(u + iv) and ty (— u + iv), the coefficients of which are
functions of u though not necessarily algebraical functions of u. Let
0 = — 2u ; and let fa, -\/r3 denote ty (w), -v/r (w + 0) respectively ; then the
equation can be represented in the form

algebraical and rational in fa and fa, but not necessarily algebraical in 9.

Because the functions allow continuous variation, we can expand ty3 in
powers of 6 : hence

When this equation, which is satisfied for all values of w and of 0, where
w and 6 are independent of one another, is arranged in powers of 6, the
coefficients of the various powers of Q must vanish separately. The coefficient
independent of 0, when equated to zero, can only lead to an identity, for it
will obviously involve only fa : any non-evanescent equation would determine
>Jr2 as a constant. Similarly, the coefficient of every power of 0, which
involves none of the derivatives of fa, must vanish identically. The co
efficient of the lowest power of 0, which does not vanish identically, involves

fa, ™2 and constants: but, because the equation g(fa, fa, 6} = 0 is

algebraical in fa, the second and higher derivatives of fa, associated with
the second and higher powers of 6 in the expansion of fa, cannot enter into
the coefficient of this power of 6. Hence we have

*(*-£)-* . . ,

F. 37

578 FAMILIES OF [280.

an algebraical equation between fa and -^2, the coefficients of which are
constants.

The coefficient of the next power of 0 will involve -^ , and so on for the

powers in succession. Instead of using the equations, obtained by making
these coefficients vanish, to deduce an algebraical equation between fa

and any one of its derivatives, we use h = 0. Thus for -^ , the equation
would be obtained by eliminating fa between the (algebraical) equations

and so for others.

Returning now to the equation

g(fa,fa,0) = 0,

in which, as it is algebraical in fa and fa, only a limited number of co
efficients, say k, are functions of 6, we can remove these coefficients as
follows. Let k—l differentiations with regard to w be effected : the resulting
equations, with g=0, are sufficient to determine these k coefficients alge
braically in terms of fa, fa and their derivatives. But the coefficients are
functions of 6 only and do not depend upon w : hence the values obtained for
them must be the same whatever value be assigned to w. Let, then, a zero
value be assigned: fa and its derivatives become constants; fa becomes
•^(0), say fa, and all its derivatives become derivatives of fa; so that the
coefficients can be algebraically expressed in terms of fa and its derivatives.
When these values are substituted in g = 0, it takes the form

$i(fa, fa, fa, fa', fa",...)=Q,

algebraical in each of the quantities involved. But between ijr, and each
of its derivatives there subsists an algebraical equation with constant co
efficients: by means of these equations, all the derivatives of fa can be
eliminated from ^ = 0, and the final form is then an algebraical equation

G (fa, fa, fa) = 0,
involving only constant coefficients. But

fa = -^ (0), fa = ^r (w), fa, = ^r (w + 0) ;

and therefore the function ty (w) possesses an algebraical addition-theorem.
Now •xjr (w) and <J> (w) are connected by the algebraical equation

therefore <f>(w) possesses an algebraical addition-theorem. But, by § 151,

280.] ALGEBRAICAL ISOTHERMAL CURVES 579

when a function </> (w) possesses an algebraical addition-theorem, it is an
algebraical function either of w, or of e*w, or of an elliptic function of w, the
various constants that arise being properly chosen : and hence the only
equations

X(z, w) = Q,

which can give families of algebraical curves in the z-plane as the conformal
equivalent of the parallel lines, u = constant, in the w-plane, are such that z
is connected by an algebraical equation either with w, or with a simply -periodic
function of w, or with a doubly -periodic function of w.

There are three sets of fundamental systems, as Schwarz calls them, of
algebraical curves determined as potential curves by a variable parameter:
they are curves such that all the others can be derived from them solely by
algebraical functions.

The first set is fundamental for the case when z is an algebraical function
of w : it is given by

u — constant,

being a series of parallel straight lines.

The second set is fundamental for the case when z is an algebraical
function of e*w ; if W denote e*w, then z is an algebraical function of W, and
all the associated curves in the .z-plane are conformal representations of the
algebraical curves in the TF-plane. If p = a + (3i, where a and /3 are real,
then

(a2 + /32) u = \ a log (X2 4- F2) + $ tan"1 ^ ,

A

a relation which can lead to algebraical curves in the W-plane only if a or
/3 be zero. If a be zero, then p is a pure imaginary, and the TF-curves are
straight lines, concurrent in the origin : if /3 be zero, then p, is real, and the
TF-curves are circles with the origin for a common centre. Hence the set
of fundamental systems for the case, when z is an algebraical function of e*w,
consists of an infinite series of concurrent straight lines and an infinite series
of concentric circles, having for their common centre the point of concurrence
of the straight lines.

The third set is fundamental for the case when z is an algebraical function
of a doubly-periodic function, say, of sn

Ex. Prove that either the modulus Tc is real or that an algebraical transformation of
argument to another elliptic function having a real modulus is possible : and shew that the
set of fundamental curves are quartics, which are the stereographic projection of confocal
sphero- conies. (Schwarz, Siebeck, Cayley.)

We thus infer that all families of algebraical curves, determined as
potential curves by a variable parameter, are conformal representations of

37—2

580 FAMILIES OF [280.

one or other of these sets of fundamental systems, by equations which are
algebraical.

But though it is thus proved that the relation between z and w must
express z as an algebraical function either of w, or of e*w, or of sn i^w, in
order that a group of algebraical curves may be the conformal representation
in the 2-plane of the lines u = constant in the w-plane, the same limitation
does not apply, if we take a single algebraical curve in the z-plane as the
conformal representation of a single line in the w-plane.

Let w = ~TIT : then the lines in the W -plane, which correspond to the
L+W

parallel lines, u = constant, in the w-plane, are the system of circles

Now consider a relation

—

7T

where Z is as yet some unspecified function of z : then

7T

WTOr ~ 7

Hence TWW0 = su{ - Z sn {--Z0 ,

k \ 7T / V 7T /

so that, if W describe the circle corresponding to u = 0, we have

7 7

r = sn (--Z sn [--Z0 ,

K \ 7T / \ 7T

whence Z — Z0 = , .

If Z = sin"1 z, and therefore Z0 = sin"1 z0, then

2iy = z — z0 = '2cos^(Z + Z0) sin -^- =i(q * — ?T) cos

so that — ^- j — h —^ f— = | ,

an ellipse, agreeing with the result in § 257, Ex. 6. This is obtained from
the relation

, ,1-w (2K . . N

= sn • — sin 1 z } ,

+w \ TT

280.] ALGEBRAICAL ISOTHERMAL CURVES 581

which is not included in the general forms of relation obtained in the
preceding investigation.

But the equation

sn t*z + - » sn Z

tt+l

does not lead to an algebraical relation between x and y for a general (non
zero) value of u. Neither the conditions of the proposition nor its limita
tions apply to this case.

The problem of determining the kinds of functional relation which will
represent a single algebraical curve in the ^-plane upon a single line of the
w-plane is wider than that which has just been discussed: it is, as yet,
unsolved.

CHAPTER XXL

GROUPS OF LINEAR SUBSTITUTIONS.

281. THE properties of the linear substitution

az + b
w = - —-,,

cz + d

considered in Chap. XIX. as bearing upon the conformal representation of two
planes, were discussed solely in connection with the geometrical relations of
the conformation : but the applications of these properties have a significance,
which is wider than their geometrical aspect.

The essential characteristic of singly-periodic functions and of doubly-
periodic functions, each with additive periodicity, is the reproduction of the
function when its argument is modified by the addition of a constant quantity.
This modification of argument, uniform and uniquely reversible, is only a
special case of a more general modification which is uniform and uniquely
reversible, viz., of the foregoing linear substitution. This substitution may
therefore be regarded as the most general expression of linear periodicity,
in a wider sense : and all functions, characterised by the property in the
general form or in special forms, may be called automorphic.

Our immediate purpose is the consideration of all the points in the
plane, which can be derived from a given point z and from one another by
making z subject to a set of linear substitutions. The set may be either
finite or infinite in number; it is supposed to contain every substitution
which can be formed by combining two or more substitutions. Such a set
is called a group.

The substitution is often denoted by S(z), or by

az + b

it is said to be in its normal form, when the real part of a (if a be a complex
constant) is positive and ad — be = 1 .

The ideas of the theory of groups of substitutions are necessary for a proper considera
tion of the properties of automorphic functions. What is contained in the present chapter
is merely sufficient for this requirement, being strictly limited to such details as arise in
connection with these special functions. Information on the fuller development of the
theory of groups, which owes its origin as a distinct branch of mathematics to Galois,

281.] FUNDAMENTAL SUBSTITUTIONS 583

will be found in appropriate treatises such as those of Serret*, Jordan t, NettoJ, and
Klein §: and in memoirs by Klein ||, Poincare**, Dyckff, and BolzaJJ. The account of
the properties of groups contained in the present chapter is based upon the works of
Klein and Poincare just quoted.

A substitution can be repeated ; a convenient symbol for representing
the substitution, that arises from n repetitions of S, is Sn. Hence the various
integral powers of S, considered in § 258, are substitutions, indicated by the
symbols tf2, S3,S*,....

But we have negative powers of S also. The definition of S°(z) is
given by

SS°(z) = S(z),

so that S° (z) = z and it is often called the identical substitution : the
definition of S~l(z) is given by

SS-l(z)=-S°(z) = z,
so that S~* (z) is a substitution the inverse of S ; in fact, if w = S (z) =

cz

then z = S~lw = -- . And then, from S~l z, by repetition we obtain
cw — a J

s-2, s-3, s-',....

If some of all the substitutions to which a variable z is subject be
not included in S and its integral powers, then we have a new substitution
T and its integral powers, positive and negative. The variable is then
subject to combinations of these substitutions : and, as two general linear
substitutions are not interchangeable, that is, we do not have T(Sz) = 8(Tz)
in general, therefore among the substitutions to which z is subject there
must occur all those of the form

where a, j3, y, 8, ... are positive or negative integers.

If, again, there be other substitutions affecting z, that are not included
among the foregoing set, let such an one be U: then there are also powers
of U and combinations of S, T, U (with integral indices) operating in any
order: and so on. The substitutions S, T, U, ... are called fundamental:
the sum of the moduli of a, /3, 7, S, ... of any substitution, compounded from
the fundamental substitutions, is called the index of that substitution ; and
the aggregate of all the substitutions, fundamental and composite, is the
group.

* Cours d'Algebre Superieure, t. ii, Sect, iv, (Paris, Gauthier-Villars).
t Traite des substitutions, (ib., 1870).

J Substitutionentheorie und ihre Anwendung auf die Algebra, (Leipzig, Teubner, 1882).
§ Vorlesungen iiber das Ikosaeder, (ib., 1884).

|| Math. Ann., t. xxi, (1883), pp. 141—218, where references to earlier memoirs by Klein are
given.

** Acta Math., t. i, (1882), pp. 1—62, pp. 193—294 ; ib., t. iii, (1883), pp. 49—92.
tt Math. Ann., t. xx, (1882), pp. 1—44, ib., t. xxii, (1883), pp. 70—108.
JJ Amer. Journ. of Math., vol. xiii, (1890), pp. 59—144.

584 CONTINUOUS AND DISCONTINUOUS [281.

There may however be relations among the substitutions of the group,
depending on the fundamental substitutions ; they are, ultimately, relations
among the fundamental substitutions, though they are not necessarily the
simplest forms of those relations. Hence, as we may have a relation of
the form

the index of a composite substitution is not a determinate quantity, being
subject to additions or subtractions of integral multiples of quantities of the
form (a) + (6) -f (c) + . . . , there being one such quantity for every relation :
we shall assume the index to be the smallest positive integer thus obtainable.

282. There are certain classifications which may initially be associated
with such groups, in view of the fact that the arguments are the arguments
of uniform automorphic functions satisfying the equation

/«&)-/(,):

in this connection, the existence of such functions will be assumed until their
explicit expressions have been obtained.

Thus a group may contain only a finite number of substitutions, that is,
the fundamental substitutions may lead, by repetitions and combinations, only
to a finite number of substitutions. Hence the fundamental substitutions,
and all their combinations, are periodic in the sense of § 260, that is, they
reproduce the variables after a finite number of repetitions.

Or a group may contain an infinite number of substitutions : these may
arise either from a finite number of fundamental substitutions, or from an
infinite number. The latter class of infinite groups will not be considered
in the present connection, for a reason that will be apparent (p. 598, note)
when we come to the graphical representations. It will therefore be
assumed that the infinite groups, which occur, arise through a finite
number of fundamental substitutions.

A group may be such as to have an infinitesimal substitution, that is,
there may be a substitution - — 3 , which gives a point infinitesimally near

to z for every value of z. It is evident there will then be other infinitesimal
substitutions in the group; such a group is said to be continuous. If there
be no infinitesimal substitution, then the group is said to be discontinuous,
or discrete.

But among discontinuous groups a division must be made. The definition
of group-discontinuity implies that there is no substitution, which gives an
infinitesimal displacement for every value of z : but there may be a number
of special points in the plane for regions in the immediate vicinity of which
there are infinitesimal displacements. Such groups are called improperly

282.] SUBSTITUTIONS 585

discontinuous in the vicinity of such points: all other groups are called
properly discontinuous. For instance, with the group of real substitutions

az + 13

8 '•'

where a, /3, 7, 8, are integers such that «S — £7 = 1, it is easy to see that, when
z-i and £2 are real, we can make the numerical magnitude of

7^ + 8 <yz.2 + 8

as small a non-evanescent quantity as we please by proper choice of a, /3,<y,8:
thus the group is improperly discontinuous, because for real values of the
variable it admits infinitesimal transformations. But such infinitesimal
transformations are not possible, when z does not lie on the axis of real
quantities, that is, when z is complex : so that, for all complex values of
0, the group is properly discontinuous.

The various points, derived from a single point by linear substitutions,
will, in subsequent investigations, be found to be arguments of a uniform
function. Continuous groups would give a succession of points infinitely
close together ; that is, for these points, we should have f (z) unaltered in
value for a line or a small area of points and therefore constant everywhere.
We shall therefore consider only discontinuous groups.

A group containing only a finite number of substitutions is easily seen to
be discontinuous : hence the groups which are to be considered in the present
connection are the discontinuous groups which arise from a finite number of
fundamental substitutions*.

The constants of all linear substitutions of the form - =. are stip

es + d

posed subject to the relation ad — be = 1. This condition holds for all
combinations, if it hold for the components of the combination. For let

OLZ

„ _

,

jz + 8 ' cz + d'

Az

whence A D - EG = (a8 - /3y) (ad - be) = 1 .

It is easy to see that ST(= U) and TS ( = V) are of the same class, that
is, they are elliptic, parabolic, hyperbolic or loxodromic together : but there is
no limitation on the class arising from the character of the component sub
stitutions.

* These discontinuous, or discrete, groups will be considered from the point of view of auto-
morphic functions. But the theory of such groups, which has many and wide applications quite
outside the range of the subject of this treatise, can be applied to other parts of our subject.
Thus it has been connected with the discussion of Eiemann's surfaces by Dyck, Math. Ann.,
t. xvii, (1880), pp. 473—509, and by Hurwitz (I.e., p. 406, note).

586 FINITE GROUPS [282.

Moreover, if U — V, so that S and T are interchangeable, then

a — d c b

that is, 8 and T have the same fixed points. They can be applied in any
order ; and, for any given number of occurrences of 8 and a given number of
occurrences of T, the composite substitution will give the same point. Thus
if 8 = z + a), then T = z + w ; if S = kz, then T = k'z. The class of func
tions, which have their argument subject to interchangeable substitutions
of the former category, have already been considered : they are the periodic
functions with additive periodicity. The group is 8™^', (—z + mw + m'a)'),
for all integral values of m and of m'.

The latter class of functions have what may be called a factorial
periodicity, that is, they resume their value when the argument is mul
tiplied by a constant*.

283. Some examples have already been given, of groups containing a
finite number of substitutions^, in the case of certain periodic elliptic
substitutions. The effect of such substitutions is (p. 514) to change a
crescent-shaped part of the plane having its angles at the (conjugate) fixed
points of the substitution into consecutive crescent-shaped parts : and so to
cover the whole plane in the passage of a substitution through the elements
constituting its period. They form the simplest discontinuous group — in
that they have only one fundamental substitution and only a finite number
of derived substitutions.

The groups which are next in point of simplicity are those with only
two substitutions that are fundamental and only a finite number that
are composite. Both of the fundamental substitutions must be periodic,
and therefore elliptic, by § 260. Taking one of these groups as an example,
one of its fundamental substitutions has + 1 as its fixed points and it is
periodic of the second order: it is evidently

w = Sz = - .
z

The other has - and oo as its fixed points, and it is periodic of the second
2i

order : it is evidently

w = Tz=\-z.

* Functions having this property are discussed in Eausenberger's Theorie der periodischen
Functionen, (Leipzig, Teubner, 1884) : in particular, in Section VI.

t The complete theory of finite groups of linear substitutions is discussed, partly in its
geometrical relation with polyhedral functions, by Klein, Math. Ann., t. ix, (1876), pp. 183 — 188,
and, in its algebraical aspect, by Gordan, Math. Ann., t. xii, (1877), pp. 23—46. A reference to
these memoirs will shew that the previous chapter contains all the essentially distinct finite
groups of linear substitutions.

283.] EXAMPLE OF INFINITE GROUP 587

Evidently &z = z, T2z = z, (S=S~\ T= T~l), so that we have already all the
powers of the fundamental substitutions taken separately.

But it is necessary to combine them. We have Uz = STz — - - , a new

1 — z

substitution : and then

Z7a*=— , U*z = z,

z

so that U is periodic of the third order. Again

z

which is not a new substitution, for Vz — U2z : and it is easy to see that there
is only one other substitution, which may be taken to be either TUz or SVz :
it gives

-,
z — 1

again periodic of the second order.

Hence the group consists of the six substitutions for z given by

1 z-\ z

i-z' z ' t-r

taking account of the identical substitution.

These finite discontinuous groups are of importance in the theory of
polyhedral functions : to some of their properties we shall return later.

Next, and as the last special illustration for the present, we form a
discontinuous group with two fundamental substitutions but containing an
infinite number of composite* substitutions. As one of the two that are
fundamental, we take

w=Tz=--,

z

which is elliptic and periodic of the second order. As the other, we take

w = Sz = z + 1,
which is parabolic and not periodic. All the substitutions are real.

Evidently T*z = z, so that T=T~l: and 8mz = z + m, where m is any
integer. Then all the composite substitutions, are either of the form
...SPTS"TSmz or of the form ...SvTSnTSmTz, both of these being included

in - — -, , where a, b, c, d are integers, such that ad - be = 1.

C£ ~T" Cv

Ex. Prove the converse — that the substitution aZ- — T , where a. b, c, d are integers

cz + d

such that ad -be — I, is compounded of the substitutions S and T.

One such group has already occurred : its fundamental (parabolic) substitutions were

588 DIVISION OF PLANE [283.

This group, again, is of the utmost importance : it arises in the theory of
the elliptic modular-functions. As with the polyhedral groups, the general
discussion of the properties will be deferred : but it is advantageous to
discuss one of its properties now, because it forms a convenient introduction
to, and illustration of, the corresponding part of the theory of groups of
general substitutions.

284. In the discussion of the functions with additive periodicity, it was
found convenient to divide the plane into an infinite number of regions such
that a region was changed into some other region when to every point of the
former was applied a transformation of the form z + mco + m'w', that is, a
substitution : and the regions were so chosen that no two homologous points,
that is, points connected by a substitution, were within one region, and each
region contained one point homologous with an assigned point in any region
of reference.

Similarly, in the case when the variable is subject to the substitutions of
an infinite group, it is convenient to divide the plane into an infinite number
of regions ; each region is to be associated with a substitution which, applied to
the points of a region of reference, gives all the points of the region, and each
region is to contain one and only one point derived from a given point by
the substitutions of the group. It is a condition that the complete plane is
to be covered once and only once by the aggregate of the regions.

When the discontinuous group has only the two fundamental substitutions,

Sz = z + I and Tz = -- , the division of the plane is easy : the difficulty of

z

determining an initial region of reference is slight, relatively to that which
has to be overcome in more general groups*.

The ordinates of z and w (= Sz) are positive together or negative together ;
and similarly for the ordinates of z and w( = Tz) : so that it will suffice to
divide the half-plane on the positive side of the axis of real quantities.

For the repetitions of the substitution S, it is evidently sufficient to divide
the plane into a series of strips, bounded by straight lines parallel to the axis
of y at unit distance apart.

For the application of the substitution T, we have to invert with regard
to a circle of radius 1 and centre the origin and to take the reflexion of the
inversion in the axis of y.

In these circumstances, we can choose as an initial region of reference, the
space bounded by the conditions

* In addition to the references already given, a memoir by Hurwitz, Math. Ann., t. xviii,
(1881), pp. 531 — 544, may be consulted for this group.

284.]

BY ELLIPTIC MODULAR-FUNCTION GROUP

589

It is sufficient to prove that any point in this region when subjected to a

substitution of the group, necessarily of the form , , where a, b, c, d are

cz+ d

integers such that ad — bc = I, is transformed to some point without the
region, and that the aggregate of the regions covers the half-plane.

If c be 0, then a= 1 = d and the transformation is only some power of S,
which transforms the point out of the region.

If c be + 1, then, since ad — be— 1, we have

1

w — a = , ,

z + d'

a and d being integers. For any point z within the region, z + d\, which is
the distance of the point from some point 0, + 1, + 2, ... on the axis of x, is
> 1 : hence

w — a\ < 1,

that is, the distance of w from some point 0, ± 1, ± 2, ... on the axis is < 1,
and therefore the transformed point is without the region.

Similarly, if c be — 1.

If cl be >1, then

As z is within the region,

so

that

all

w = — -
c c - ^ ^ d

d
c

c

^ -^- : and therefore
Z

V3 a ^ I ^ 1

2 W c < c2 < 4 '

w; —
c

.-. 1

Hence the distance of w from some point of the axis is < | ^3, that is, the
transformed point is without the region.

The exceptions are points on the boundary of the region. The boundary
x = -\ is transformed by S to x = + % : the boundary #2 + 7/2 = 1 is trans
formed by T into itself: but all other points are transformed into others
without the region.

We now apply the substitutions 8 and T to this region and to the
resulting regions. Each substitution is uniform and is reversible: so that
to a given point in the initial region there is one, and only one, point in each
other region.

The accompanying diagram (Fig. 108) gives part of the division of the
plane into regions, the substitutions associated with each region being
placed in the region in the figure ; it is easy to see that the aggregate of
regions completely covers the half-plane. All the linear boundaries of S",
for different integral values of n, are changed by the substitution T into
circles having their centres on the axis of x and touching at A : thus the
boundary between 8 and S1 is transformed into the boundary between

590

ELLIPTIC MODULAR-FUNCTION GROUP

[284.

TS and TS2. All the lines which bound the regions are circles having
their centres on the axis of x or are straight lines perpendicular to that
axis; and the configuration of each strip is the same throughout the
diagram.

Fig. 108.

It will be noticed that in one region there are two symbols, viz., S~1TS~*
and TST : the region can be constructed either by $~] applied to TS~l or by
T applied to ST. It therefore follows that

TST=S-1T8-\

Hence 8 . TST . S = S . S^ TS~* . S = T,

or, since T2 = 1, we have STSTST = 1 = TSTSTS,

a relation among the fundamental substitutions. Thus the symbol of any
region is not unique : and, as a matter of fact, if we pass clockwise in a small
circuit round 0 from the initial region, we find the regions to be 1, T, TS, TST,
TSTS, TSTST, TSTSTS, the seventh being the same as the first and giving
the above relation.

By means of this relation it will be found possible to identify the non-
unique significations of the various regions. At each point there are six
regions thus circulating always, either in the form ®S, SST, ®STS, ... or in
the form ® T, ®TS, ®TST, .... And by successive transformations, the space
towards the axis of x is distributed into regions.

The decision of the region to which a boundary should be assigned will
be made later in the general investigation ; it will prove a convenient step
towards the grouping of edges of a region in conjugate pairs.

284.] FUCHSIAN GROUPS

Note. It may be proved in the same way that, for any discontinuous
group of substitutions, the plane of the variable can be divided into regions
of a similar character. As will subsequently appear, there is considerable
freedom of choice of an initial region of reference, which may be called a
fundamental region.

285. We pass now to the consideration of the more general discontinuous
groups, based on the composition of a finite number of fundamental substitu
tions. By means of these groups and in connection with them, the plane of
the variable can be divided into regions, one corresponding to each substitu
tion of the group. The regions are said to be congruent to one another :
the infinite series of points, one in each of the congruent regions, which arise
from z when all the substitutions of the group are applied to z, are said to
be corresponding or homologous points : and the point in .R0 of the series is
the irreducible point of the series. As remarked before, the correspondence
between two regions is uniform : interiors transform to interiors, boundaries
to boundaries.

Two regions 'are said* to be contiguous, when a part of their boundaries is
common to both. Each region, lying entirely in the finite part of the plane,
is closed : the boundary is made up of a succession of lines which may for
convenience be called edges, and the meeting-point of two edges may for con
venience be called a corner.

Such a group, when all the substitutions are real, is called^ Fuchsian,
by Poincare ; the preceding example will furnish a simple illustration, useful
for occasional reference. All the substitutions are of the form

asz + b,

csz + ds '

which form will be denoted by fs (z}. We shall suppose that an infinite
group of real substitutions is given, and that it is known independently to
be a discontinuous group: we proceed to consider the characteristic properties
of the associated division of the plane, which is to be covered once and only
once by the aggregate of the regions. The fundamental region is denoted
by _R0: the region, which results when the substitution fm(z) is applied
to the points of ^0, will be denoted by Rm.

So long as we deal with real substitutions, it is sufficient to divide the
half-plane above the axis of x into regions : and this axis may be looked upon
as a boundary of the plane. Since the group is infinite, the division into
regions must extend in all directions in the plane to its finite or infinite
boundaries : for we should otherwise have infinitesimal transformations. Thus

* Poincare uses the term limitroplies.

t Math. Ann., t. xix, p. 554, t. xx, pp. 52, 53 : Ada Math., t. i, p. 62. The same term is
applied to a less limited class of groups; see p. GOG, note.

592 CATEGORIES [285.

the edge of a region is either the edge of a contiguous region, and then it is
said to be of the first kind ; or it is a part of the boundary of the plane, that
is, in the present case it is a part of the axis of x : and then it is said to be of
the second kind. Since all real substitutions transform a point above the axis
of x into another point above the axis of x, it follows that all edges congruent
with an edge of the first kind (an edge lying off the axis of x) themselves
lie off the axis of x, that is, are of the first kind : and similarly all edges con
gruent with an edge of the second kind are themselves of the second kind.

The corners, being the extremities of the edges, are of three categories.
If a corner be an extremity of two edges of the first kind and not on the
axis of x, then it is of the first category : and the infinite series of corners
homologous with it are of the first category. If it be common to two
edges of the first kind and lie on the axis of x, then it is of the second
category: and the infinite series of corners homologous with it are of the
second category. If it be common to two edges, one of the first and one of
the second kind, it is of the third category ; of course it lies on the axis of
x and the infinite series of corners homologous with it are of the third
category. We do not consider two edges of the second kind as meeting :
they would, in such a case, be regarded as a single edge.

Each edge of the first kind belongs to two regions. We do not assign
such an edge to either of the regions, but we use this community of
region to range edges as follows. Let the edge be Ep, common to R0
and Rp ; then, making the substitution inverse to fp (z), say fp~l (z), Rp
becomes R0, R0 becomes R-p, and Ep becomes fp~l (Ep), which is necessarily
an edge of the first kind and is common to the new regions R_p and RQ,
that is, it is an edge of R0. Let it be Ep' : then Ep and Ep' may be
the same or they may be different.

If Ep and Ep be different, then we have a pair of edges congruent to
one another : two such congruent edges of the same region are said to be
conjugate. Since the substitutions are of the linear type, the correspondence
being uniform, not more than one edge of a region can be conjugate with
a given edge of that region.

If Ep and Ep' be the same, then the substitution transforms Ep into
itself : hence some point on Ep must be transformed into itself. As the edge
is of the first kind so that the point is above the axis of X, the substitution
is elliptic and has this point as the fixed point of the substitution in
the positive half-plane. The two parts of Ep can be regarded as two
edges : and the common point as the corner, evidently of the first category.
Because the directions of the edges measured away from the point are
inclined at an angle TT, it follows that the multiplier of the elliptic sub
stitution is eni, or — 1. An illustration of this occurs in the special
example of § 284, where the circular boundary of the initial region of

285.] FUNDAMENTAL SUBSTITUTIONS 593

referenc

that is,

reference is changed into itself by the fundamental substitution wz = — 1,

w — i z — i

w + i z + i

Hence the edges of the first kind are even in number and can be arranged
in conjugate pairs.

Further, a point on an edge of the first kind is transformed into a
point on the conjugate edge — uniquely, unless the point be a corner, when
it belongs to two edges. Hence points on edges of the first kind other than
corners correspond in pairs.

An edge of the second kind is transformed into one of the second kind,
but belonging to a different polygon : there is no correspondence between
points on edges of the second kind belonging to the same polygon.

Each corner, as the point common to two edges, belongs to at least three
regions. As a point of one edge, it will have as its homologue an extremity
of the conjugate edge : as a point of another edge, it will have as its homologue
an extremity of the edge conjugate to that other : and these homologues may
be the same or they may be different. Hence several corners of a given
region may be homologous : the set of homologous corners of a given region is
called a cycle. Since points of a series homologous with a given point all
belong to one category, it is convenient to arrange the cycles in connection
with the categories of the component points.

The number of edges of the first kind is even, say 2n : and they can be
arranged in pairs of conjugates, say E1} En+l ; E2, En+z ; ... . Then since En+p
is the conjugate of Ep, a,ndfn+p (z) is the substitution which changes R0 into
Rn+p,fn+p(z) is a substitution changing Ep into En+p. After the preceding
explanation,^"1 (z) is also a substitution changing Ep into its conjugate En+p :
hence we have

fn+p 0) =fp~l (4

Hence for a division of the plane, each region of which has 2w edges of the
first kind, the group contains n fundamental substitutions : the remaining n
substitutions, necessary to construct the remaining contiguous regions, are
obtained by taking the first inverses of the fundamental substitutions.

The edge Ep has been taken as the edge common to _R0 and Rp, the region
derived from R0 by the substitution fp (z). Every region will have an edge
congruent to Ep : if JR; be one such region, then the region, on the other side
of that line and having that line for an edge (the edge is, for that other
region, the congruent of the conjugate of Ep), is obtainable from R0 by the
substitution fi{fp(z)}. We thus have an easy method of determining the
substitution to be associated with the region, by considering the edges which
are crossed in passing to the region : and, conversely, when the substitutions
are associated with the regions, the correspondence of the edges is known.

As in the special example, there are relations among the fundamental
substitutions. The simplest mode of determining them is to describe a small
F. 38

594 CONVEXITY [285.

circuit round each corner of .R0 in succession : in the description of the circuit,
the symbol of each new region can be derived by a knowledge of the edge last
crossed and when the circuit is closed the last symbol is the symbol also of E0,
so that a relation is obtained.

286. The only limitations as yet assigned to the initial region (and there
fore to each of the regions) of the plane are (i) that it contains only one point
homologous with z, and (ii) that the even number of edges of the first kind
can be arranged in congruent conjugate pairs. But now,
without detracting from the generality of the division, we
can modify the initial region in such a way that all the
edges of the first kind are arcs of circles with their centres
on the axis of x. For let C...AFB...DGG be a region with
CGD and AFB for conjugate edges; join CD by an arc of
a circle CED with its centre on the axis of x : and apply to
CED the substitution inverse to that which gives the region
in which E lies : let AHB be the result, being also (§ 258)
an arc of a circle with its centre on the axis of x. Then the part AFBHA,
say S0, is transformed to CGD EC, say $</, by the substitution which causes a
passage from R0 across CGD into another region: every point in S0 has a
homologue in S0' : and there is, by the hypothesis that R0 is the initial region,
no homologue in R0 of a point in S0 except the point itself. If, then, we take
away 80 from R0 and add $„', we have a new region

RQ =R0 + S0' — 8Q.

It satisfies all the conditions which apply to the regions so far obtained : there
is no point in R0' homologous with a point in it, and the conjugate edges
CGD and AFB are replaced by conjugate edges CED, AHB congruent]
by the same substitution as the former pair. And the new conjugate
edges are circles having their centres on the axis of x.

Proceeding in this way with each pair of conjugate edges that are not
arcs of circles having their centres on the axis of x, and replacing it by a pair
of conjugate edges congruent by the same substitution and consisting of
arcs of circles having their centres on the axis of x, we ultimately obtain a
region in which all the edges of the first kind are arcs of circles having their
centres on the axis of x. These can, of course, be arranged in conjugate pairs,
congruent by the assigned fundamental substitutions. Straight lines perpen
dicular to the axis of x count as circles with centres at x = oo on that
axis : all other straight lines, not being parts of the axis of x, can be replaced
by circles.

The edges of the second kind are left unaltered.

A region, thus bounded, is called a normal polygon.

Further, this normal polygon may be taken convex, that is, edges do not
cross one another. If the preceding reduction of a region to the form of

286.] OF NORMAL POLYGON 595

a normal polygon should lead to a cross polygon, then, as is usual in
dealing with the area of such cross figures, part of the area is to be
considered negative: and therefore, for every point in this negative part,
there must be two points in the positive part. Hence,
in the positive part, there are

(i) points, none of which has a homologue in

the negative part, or in the positive

part except itself : their aggregate gives

a normal polygon Q :

(ii) two sets of points, each set of which consists
of the homologues of points in the nega
tive part, and makes up a positive normal
polygon; let the polygons be T, and T2. Fi8- no-

The negative part is a normal polygon T, to which T, and T2 are each congruent.
We now change R by adding a normal polygon T and subtracting a
normal polygon T, : thus for the new region we have a positive (that is, a
convex) polygon Q, and a positive (convex) polygon T2. No point in Q has a
homologue in T,: hence Ta and Q together make up a region such that
homologues of all points within it lie outside: this region is a normal
polygon, and it is convex. Hence we may take as the initial region of
reference a normal convex polygon, that is, a convex polygon bounded by arcs of
circles having their centres on the axis of x, or by portions of the axis ofx: the
number of arc-edges is even, and they can be arranged in conjugate pairs.

Simplicity is obtained by securing that the curves, which compose the
boundary, are as like one another in character as possible. The substitutions
are linear and they change boundaries into boundaries : the whole plane is to
be covered : and there are no gaps between a bounding edge and the homo
logue of the conjugate bounding edge. The only curves, which satisfy this
condition of leaving no gaps, and which are of the same character after any
number of linear transformations, .are circles and straight lines.

287. We have seen that two (or more than two) corners of a convex
polygon may be homologous: it is
now necessary to arrange all the

corners in their cycles. Let AB and A E

ED be two conjugate edges of a

normal polygon, and let be

cz + d . .

the substitution which changes AB c C'

into ED ; then, as usual, we have Fig< 11L

ad — be 1 11

c2 ~~d ~ c2 d '
z + - z 4- -

c c

38—2

596 CLOSED AND OPEN [287'

f a\ f d\

so that arg. [w I + arg. I z + - 1 = TT.

\ C J \ C /

This at once shews that, whatever be the value of - and of - , the points A,

c c

E are homologous, and likewise the points B, D. Hence to obtain a corner
homologous to a given corner we start from the corner, describe the edge of
the polygon beginning there, then describe in the same direction* the conju
gate edge : the extremity of that edge is a homologous corner.

The process may now be reapplied, beginning with the last point ; and it
can be continued, each stage adding one point to the cycle, until we either
return to the initial point or until we are met by an edge of the second kind.
In the former case we have a completed cycle, which may be regarded as a
closed cycle. In the latter case we can proceed no further, as edges of the
second kind are not ranged in conjugate pairs ; but, resuming at the initial
point we apply the process with a description in the reverse direction until
we again arrive at an edge of the second kind : again we have a cycle, which
may be regarded as an open cycle.

In the case of a closed cycle, if one of the included points be of the first
category, then all the points are of the first category : the cycle itself is then
said to be of the first category. If one of the points be of the second category,
then since no edge of the second kind is met in the description, all the edges
met are of the first kind ; and therefore all the points, lying on the axis of x
and being the intersections of edges of the first kind, are of the second
category : the cycle itself is then said to be of the second category.

Open cycles will contain points of the third category : they may also
contain points of the second category for points both of the second and of
the third categories lie on the axis of x, and homology of the points does not
imply conjugacy of all edges of which they are extremities. Such cycles are
said to be of the third category.

It thus appears that the cycles can be derived when the arrangement in
conjugate pairs of edges of the first kind is given ; and it is easy to see that
the number of open cycles is equal to the number of edges of the second
kind.

We may take one or two examples. For a quadrilateral, in which
the conjugate pairs are 1,4; 2, 3 — the numbers being
as in the figure — we have by the above process A, AB,
DA, A: that is, A is a cycle by itself. Then B, BC, CD,
D, DA, AB, B : that is, B and D form a cycle ; and then
C, CD, BC, C, that is, C is a cycle by itself. The cycles
are therefore three, namely, A ; B, D; C.

* This is necessary : the direction is easily settled for a complete polygon the sides of which
are described in positive or in negative direction throughout.

287.]

CYCLES

597

For a hexagon, in which the conjugate pairs are 1,5; 2, 4 ; 3, 6, the
cycles are two, namely, A, F, D, C and B, E. If the conjugate pairs be

Fig. 113.

1, 6; 2, 5; 3, 4, the cycles are four, namely, A ; B, F ; C, E ; D. If the
conjugate pairs be 1, 4 ; 2, 5 ; 3, 6 the cycles are two, namely, A, C, E;
B, D, F.

For a pentagon, with one edge of the second kind as in the figure and

having 1, 3; 4, 5 as the conjugate pairs, the cycles are three, namely,
E] A, D; B, C; the last being open and of the third category.

For a quadrilateral as in the figure, having three corners on the axis of x
and 1, 2; 3, 4 as the arrangement of its conjugate
pairs, the cycles are D\ A, C ; B; the last two
being of the second category.

We have now to consider the angles of the
polygons taken internally. It is evident that at
any corner of the second category, the angle is
zero, for it is the angle between two circles meeting
on their line of centres ; and that at any corner of
the third category the angle is right. There therefore remain only the
angles at corners of the first category. Let Al} A.2, ..., An be the corners
in a cycle of the first category and denote the angles by the same letters.

598 ANGLES IN A CYCLE [287.

Since Al and A2 are homologous corners, they are extremities of conjugate
edges. Apply to the plane, in the vicinity of A2, the substitution which
changes the edge ending in A2 to its conjugate ending in A1: then the
point A2 is transferred to the point Al\ one edge at A2 coincides with its
conjugate at Al and the other edge at A2 makes an angle
A2 with it, because of the substitution which conserves
angles. The latter edge was the edge which followed A.2
in the cycle for the derivation of As: we take its conju
gate ending in A3, and treat these and the points A2 and lg' 116'

A3 as before for A± and A2 and their conjugate edges, namely, by using the
substitutions transforming conjugate edges and passing from A3 to A., and
then those from A2 to Alf

Proceeding in this way round the cycle, we shall have

(1) a series of lines at the point, each line between two angles being

one of the conjugate edges on which the two corners lie :

(2) the angles corresponding to the corners taken in cyclical order.

Hence after n such operations we shall again reach an angle Al. If the edge
do not coincide with the first edge, we repeat the set of n operations : and
so on.

Now all these substitutions lead to the construction of the various regions
meeting in A, which are to occupy all the plane round A, and no two of
which are to contain a point which does not lie on an edge. Hence
after the completion of some set of operations, say the pih set, the
edges of A1 will coincide with their edges of the first angle A+ ; and
therefore

2-Tr

so that A! + A» + ... +An= — .

P

Hence the sum of the angles at the corners, in a cycle of the first category,
is a submultiple of 2?r.

Further, if q be the number of polygons at A, we have

np = q.

COROLLARY 1. For a cycle of the second category — it is a closed cycle —
both p and q are infinite.

The cycle contains only a finite number of corners, because the polygon
has only a finite number* of edges : as each corner is of the second category,

* If the number be infinite, the edges must be infinitesimal in length unless the perimeter of
each of the polygons is infinite : each of these alternatives is excluded.

The reason for finiteness (§ 282) in the number of fundamental substitutions in the group
is now obvious : their number is one-half of the number of edges of the first kind.

287.] OF THE FIRST CATEGORY 599

the angle is zero : and therefore the repetition of the set of operations can be
effected without limit. Hence p is infinite ; and, as n polygons at a corner
are given by each set of operations, the number q of polygons is infinite.

COROLLARY 2. Corresponding to every cycle of the first category, there is
a relation among the fundamental substitutions of the group.

Let/12 be the substitution interchanging the conjugate edges through Al
and A., ; f.^ the substitution interchanging the conjugate edges through A2
and A and so on. Let U denote

then U* 0) = z.

For U is the substitution which reproduces the polygon with the angle
A-L at Al ; and this substitution is easily seen, after the preceding explanation,
to be periodic of order p. Moreover, this substitution U is elliptic.

288. The following characteristics of the fundamental region have now
been obtained :

(i) It is a convex polygon, the edges of which are either arcs
of circles with their centres on the axis of x or are portions
of the axis of x :

(ii) The edges of the former kind are even in number and can be
arranged in conjugate pairs : there is a substitution for which
the edges of a conjugate pair are congruent; if this sub
stitution change one edge a of the pair into a', it changes
the given region into the region on the other side of a :

(iii) The corners of the polygon can be arranged in cycles of one or
other of three categories :

(iv) The angles at corners in a cycle of the second category are zero :
each of the angles at corners in a cycle of the third category
is right : the sum of the angles at corners in a cycle of the
first category is a submultiple of 27r.

Let there be an infinite discontinuous group of substitutions, such that its
fundamental substitutions are characterised by the occurrence of the fore
going properties in the edges and the angles of the geometrically associated
region : and let the whole group of substitutions be applied to the region.

Then the half-plane on the positive side of the axis of x is covered : no
part is covered more than once, and no part is unassigned to regions. It is
easy to see in a general way how this given condition is satisfied by the
various properties of the regions. Since the edges of the first kind in
the initial region can be arranged in conjugate pairs, it is so with those
edges in every region : and the substitution, which makes them congruent,

600 FUNDAMENTAL SUBSTITUTIONS [288.

makes one of them to coincide with the homologue of the other for the
neighbouring region, so that no part is unassigned. No part is covered
twice, for the initial region is a normal convex polygon and therefore every
region is a normal convex polygon : the edges are homologous from region to
region, and form a common boundary. The angle of intersection with a
given arc is sufficient to fix the edge of the consecutive polygon : for an arc
of a circle, making on one side an assigned angle with a given arc and having
its centre on the axis, is unique. At every corner of any polygon, there will
be a number of polygons : the corners which coincide there are, for the
different polygons, the corners homologous with a cycle in the original
region: and the angles belonging to those corners fill up, either alone or
after an exact number of repetitions, the full angle round the point.

We have seen that the substitution, which passes from a polygon at a
point to the same polygon, after n polygons, reproduces the angular point
at the same time as it reproduces the polygon ; the point is a fixed point
of an elliptic substitution. Similarly, if the point belong to a cycle of the
second category, n is infinite and the substitution does not change the point,
which is therefore a fixed point of the substitution ; as the fixed point is on
the axis, the substitution is parabolic (§ 292).

The preceding are the essential properties of the regions, which are
sufficient for the division of the half-plane when a group is given, and
therefore by reflexion through the axis of x, they are sufficient for the
division of the other half-plane.

The position of corners of the first category, and the orientation of edges
meeting in those corners, are determinate when the group is supposed
given : within certain limits, half of the corners of the third category can
be arbitrarily chosen.

289. In the preceding investigation, the group has been supposed given :
the problem was the appropriate division of the plane. The converse problem
occurs when a fundamental region, with properties appropriate for the
division of the half-plane, is given: it is the determination of the group.
The fundamental substitutions of the group are those which transform an
edge into its conjugate, and they are to be real — conditions which, by
§ 258, are sufficient for their construction. The whole group of substitu
tions is obtained by combining those that are fundamental. The complete
division of the half-plane is effected, by applying to each polygon in suc
cession the series of fundamental substitutions and of their first inverses.

It is evident that a given division of the plane into regions determines
the group uniquely: but, as has already been seen in the general ex
planation, the existence of a group with the requisite properties does not
imply a unique division of the plane.

289.]

EXAMPLE

601

As an example, let the fundamental substitutions be required when a quadrilateral as
in Fig. 112, having 1, 2; 3, 4 for the conjugate pairs of edges, is given as a fundamental
region. The cycles of the corners are B • D; A, C ; so that

= 3, A = C ; the

where I, m, n are integers.

The simplest case has already been treated, § 284 : there, 1 = 2,
region is a triangle, really a quadrilateral with two
edges as conterminous arcs of the same circle. We
shall therefore suppose this case excluded ; we take the
case next in point of simplicity, viz. 1=2, A = C. Then
AB and BC are conterminous arcs of one circle : we
shall take the centre of this circle to be the origin, its
radius unity and B on the axis of y ; then B is a fixed
point of the substitution, which changes AB into BC.
The substitution is

__1
z '

it is one of the two fundamental substitutions.

Fig. 117.

Evidently A = - , ADB= — . Let E be the centre of the circle AD, and p its radius :

7T
I

then OAE=-, ODE=?- - - , and so

whence

- 2p cos - = OE* = p2 cos2 - ,
'

/ \ i

• Q 7* 7T / of •*•*»*

p sin'' — = cos - + cos2 — sin2 — I ,
m n \ n m)

the negative sign of the radical corresponding to the case when D lies below ABC. The
radius p must be real and therefore

1 1

we omit the case of »i = oo, and therefore »>2.

The fundamental substitution, which changes AD into CD, has D and the complex con

jugate to D for its fixed points: these points are+^psin — . The argument of the

2
multiplier is — , being the angle ADC : hence the substitution is

W - In sin — 2 — in sin - 2irf

m m m

^^— — = ^^— — e

which reduces to

. . .

sin — 2 + ^p sin —
m m

. • o "

2 cos +p sin1*
m m

I . n '

h cos -

p m

where p has the value given by the above equation.

602 EXAMPLE [289.

This substitution, and the substitution w= — , are the fundamental substitutions of

z

the group. The special illustration in § 284 gives

m = oo, p = oo. n = 3. p sin2 — = 2 cos - = 1 ;
in n

the special form therefore is

Tciking cos-=a, cos- = &, A = («2 + 62- 1)J, we have p (I -«2) = Z> + A; the second fun-

7?2/ 72-

damental substitution is

„ az + A + b

W = SZ =

It is easy to see that

T2=l, &*=!, (TS)* = l,
gure can be co
An interesting figure occurs for m = 4, n = 6.

where Tz= — ; the complete figure can be constructed as in § 284.

In the same way it may be proved that, if an elliptic substitution have re* * for its
common points and 20 for the argument of its multiplier, its expression is

Az + B

sin (6 - 0) „ sin 9 1 sin e „ sin (6 + e)

where A = — ^ — ^— - , fl*»r-r— 3, (/=-- — — -^ , D= — J — - — -.

sin 6 sin Q r sin 6 sin 6

Taking now the more general case where B=-j-, D= — , A + C= — , let B (in

figure 112) be the point be^\ and A the point aeat. Then the substitution which transforms
AB into ^C'is the above, when 6 = ft, r = b, Q = B, so that, if C be ceyt,

-|sin Be

giving two relations among the constants.

Similarly two more relations will arise out of the substitution which transforms CD
into DA. And three relations are given by the conditions that the sum of the angles at
A and C is an aliquot part of 2?r, and that each of the angles B and D is an aliquot part

Of 27T.

290. All the substitutions hitherto considered have been real : we now
pass to the consideration of those which have complex coefficients. Let

J2 + 8

be such an one, supposed discontinuous: then the effect on a point is obtained
by displacing the origin, inverting with respect to the new position, reflecting
through a line inclined to the axis of as at some angle,, and again displacing
the origin. The displacements of the origins do not alter the character of
relations of points, lines and curves : so that the essential parts of the
transformation are an inversion and a reflexion.

290.] FUNDAMENTAL CIRCLE

Let a group of real substitutions of the character considered in the
preceding sections be transformed by the foregoing single complex substitu
tion : a new group

«• — s + b

yz + 8 ' az + 8
c — -Z
yz + 8

will thus be derived. The geometrical representation is obtained through
transforming the old geometrical representation by the substitution

(az + 13
\yz+8'

so that the new group is discontinuous.

The original group left the axis of x unchanged, that is, the line Z =
was unchanged ; hence the substitutions

ous + /3

yz + 8
will leave unchanged the line which is congruent with z = z0 by the

substitution ( - — £. z}. This line is
\yz + 8 )

yz-a. y
or it may be taken in the form

imaginary part of — - — — = 0.

<yz — a

It is a circle, being the inverse of a line ; it is unaltered by the substitutions
of the new group, and it is therefore called* the fundamental circle of this
group. The group is still called Fuchsian (p. 606, note).

The half-planes on the two sides of the axis of x are transformed into the
two parts of the plane which lie within and without the fundamental circle
respectively : let the positive half-plane be transformed into the part within
the circle.

With the group of real substitutions, points lying above the axis of x
are transformed into points also lying above the axis of x, and points below
into points below: hence with the new group, points within the fundamental
circle are transformed into points also within the circle, and points without
into points without.

* Klein uses the word Hauptkreis.

604-

GROUPS CONSERVING

[290.

The division of the half-plane into curvilinear polygons is changed into a
division of the part within the circle into curvilinear polygons. The sides of
the polygons either are circles having their centres on the axis of x, that is,
cutting the axis orthogonally, or they are parts of the axis of x : hence the
sides of the polygons in the division of the circle either are arcs of circles
cutting the fundamental circle orthogonally or they are arcs of the funda
mental circle.

The division of the part of the plane without the circle is the trans
formation of the half-plane below the axis of x, which is a mere reflexion
in the axis of x of the half-plane above : thus the division is characterised by
the same properties as characterise the division of the part within the
fundamental circle. But when the division of the part within the circle
is given, the actual division of the part without it can be more easily
obtained by inversion with the centre of the fundamental circle as centre
and its radius as radius of inversion.

This process is justified by the proposition that conjugate complexes are

transformed by the substitution ( - s , z ) into points which are the in-

\yz + 8 J

verses of one another with regard to the fundamental circle. For a system
of circles can be drawn through two conjugate complexes, cutting the real
axis orthogonally : when the transformation is applied, we have a system of
circles, orthogonal to the fundamental circle and passing through the two
corresponding points. The latter are therefore inverses with regard to
the fundamental circle.

This proposition can also be proved in the following elementary manner.

Let OC, the axis of x, be inverted, with A as the centre of inversion, into a circle :
P and Q be two conjugate complexes, and
let AP cut axis of x in C: let CQ cut the
diameter of the circle in R. Since OC bisects
PQ, it bisects AR; and therefore the centre
of the circle is the inverse of R.

Let p and q be the inverses of P and Q :
joinpq,qr. Then th e angle pq Q = CPQ = CQP,
and Aqr = CRO: ihus pqr is a straight line.
Also

_qr _QR_AP _Ar

1q~~AR~jR~Ap'

pr _PR_AQ_Ar

Ap~ AR~~AR~^ Fig. 118.

so that rp.rq = Ar2.

Thus p and q are inverses of each other, relative to r and with the radius of the funda
mental circle as radius. Transference of origin and reflexion in a straight line do not alter
these properties : and therefore^ and q, the transformations of the conjugate P and §, are
inverses of one another with regard to the fundamental circle.

290.] A FUNDAMENTAL CIRCLE 605

Hence with the present group, constructed from an infinite discontinuous
group of real substitutions transformed by a single complex substitution, the
fundamental circle has the same importance as the axis of real quantities
in the group of real substitutions. It is of finite radius, which will be taken
to be unity : its centre will be taken to be the origin. The area within it is
divided into regions congruent with one another by the substitutions of the
group : the whole of the area is covered by the polygons, but no part is
covered more than once.

All the points, homologous with a given point z within the circle, lie
within the circle : each polygon contains only one of such a set of homologous
points.

The angular points of a polygon can be arranged in cycles which are
of three categories. The sum of the angles at points in a cycle of the first
category is unchanged by the substitution ; it is equal to an aliquot part of
2?r. At points in a cycle of the second category each angle is zero : at points
in a cycle of the third category each angle is right.

In fact, all the properties obtained for the division of the plane into
polygons now hold for the division of the circle into polygons associated
with the group

as + /3 7.2 + 8

provided we make the changes that are consequent on the transformation of
the axis of x into the fundamental circle.
The form of the substitution

which secures that the fundamental circle in the w-plane shall be of radius unity and
centre the origin, is easily obtained.

It has been proved that inverse points with respect to the circle correspond to conjugate
complexes; hence w = 0 and w=oo correspond to two conjugate complexes, say X and X0,
and therefore

z-\

W = K r- ,

z-V

where | K \ = 1 because the radius of the fundamental circle is to be unity. The presence of
this factor *c is equivalent to a rotation of the w-plane about the origin. As the origin
is the centre of the fundamental circle, the circle is unaltered by such a change :
and therefore, without affecting the generality of the substitution, we may take « = !,
so that now

where X is an arbitrary complex constant. The substitution is not in its canonical form,
which however can at once be deduced.

291. It has been seen, in § 260, that, when any real substitution is para
bolic or hyperbolic, then practically an infinite number of points coincide with

606 FUCHSIAN GROUPS [291.

the fixed point of the substitution when it is repeated indefinitely, whatever
be the point z initially subjected to the transformation ; this fixed point lies
on the axis of x, and is called an essential singularity of the substitution.
When we consider such points in reference to automorphic functions, which
are such as to resume their value when their argument is subjected to
the linear substitutions of the group, then at such a point the function
resumes the value which it had at the point initially transformed ; that is,
in the immediate vicinity of such a fixed point of the substitution, the
function acquires any number of different values : such a point is an essential
singularity of the function. Hence the essential singularities of the group
are the essential singularities of the corresponding function.

Now all the essential singularities of a discontinuous group lie on the
axis of x when the group is real ; the line may be or may not be a con
tinuous line of essential singularity. If, for example, x be any such point
for the group of §§ 283, 284 which is characteristic of elliptic modular-
functions, then all the others for that group are given by

ax + b

ex + d'

where a, b, c, d are integers, subject to the condition ad — be = 1 : and
therefore all the essential singularities are given by rational linear trans
formations. For points on the real axis, this group is improperly dis
continuous : and therefore for this group the axis of x is a continuous line
of essential singularity.

Hence when we use the transformation ( ~ , z } to deduce the division

\yz + 8 )

of the fundamental circle into regions, the essential singularities of the new
group are points on the circumference of the fundamental circle : the cir
cumference is or is not a continuous line of essential singularity for the
function or the group*, according as the group is properly or improperly
discontinuous for the circle.

292. It is convenient to divide the groups into families, the discrimin
ation adopted by Poincare being made according to the categories of cycles of
angular points in the polygons into which the group divides the plane. The
group is of the

1st family, if the polygon have cycles of the 1st category only,

2nd 2nd ,

3rd 3rd ,

4th 2nd and 3rd ,

* Poincar6 calls the group Fuchsian, both when all the coefficients are real and when they
arise from the transformation of such an infinite group by a single substitution that has imaginary
coefficients. A convenient resume of his results is given by him in a paper, Math. Ann., t. xix,
(1882), pp. 553—564.

292.] FAMILIES OF GROUPS 607

5th family, if the polygon have cycles of the 1st and 3rd categories only,

6th 1st and 2nd ,

7th all three categories.

Thus in the polygons, associated with groups of the 1st, the 2nd, and the 6th
families, all the edges are of the first kind ; in the polygons associated with
groups of the remaining families, edges of the second kind occur.

A subdivision of some of the families is possible. It has been proved that
the sum of the angles in a cycle of the first category is a submultiple of 2vr.
If the sum is actually 2?r, the cycle is said to belong to the first sub-category:
if it be less than 2?r (being necessarily a submultiple), the cycle is said to
belong to the second sub-category. And then, if all the cycles of the polygon
belong to the first sub-category, the group is said to belong to the first order
in the first family: if the polygon have any cycle belonging to the second
sub-category, the group is said to belong to the second order in the first
family.

It has been proved in § 288 that a corner belonging to a cycle of the
second category is not changed by the substitution which gives the conti
guous polygons in succession ; the corner is a fixed point of the substitution,
so that the substitution is either parabolic or hyperbolic. In his arrange
ment of families, Poincare' divided the cycles of the second category into
cycles of two sub-categories, according as the substitution is parabolic or
hyperbolic : but Klein has proved* that there are no cycles for hyperbolic
substitutions, and therefore the division is unnecessary. The families of
groups, the polygons associated with which have cycles of the second
category, are the second, the fourth, the fifth and the seventh.

There is one very marked difference between the set of families, con
sisting of the first, the second and the sixth, and the set constituted by
the remainder.

No polygon associated with a real group in the former set has an edge of
the second kind : and therefore the only points on the axis taken account of
in the division of the plane are the essential singularities of the group.
The domain of any ordinary point on the axis in the vicinity of each of the
essential singularities is infinitesimal : and therefore the axis of x is taken
account of in the division of the plane only in so far as it contains essential
singularities of the group and the functions. This, of course, applies equally
to the transformed configuration in which the conserved line is the funda
mental circle : and therefore, in the division of the area of the circle, its
circumference is taken account of only in so far as it contains essential
singularities of the groups and the functions.

But each polygon associated with a real group in the second set of
families has an edge of the second kind : the groups still have all their

* Math. Ann., t. xl, (1892), p. 132.

608 CLASS OF GROUPS [292.

essential singularities on the axis of x (or on the fundamental circle)
and at least some of them are isolated points ; so that the domain of an
ordinary point on the axis is not infinitesimal. Hence parts of the axis of
x (or of the circumference of the fundamental circle) fall into the division of
the bounded space.

293. There is a method of ranging groups which is of importance in
connection with the automorphic functions determined by them.

The upper half of the plane of representation has been divided into
curvilinear polygons ; it is evident that the reflexion of the division, in the
axis of real quantities, is the division of the lower half of the plane. Let the
polygon of reference in the upper half be R0 and in the lower half be R0',
obtained from R0 by reflexion in the axis of real quantities. Then, if the
group belong to the set, which includes the first, the second and the sixth
families, _R0 and ^Ro' do not meet except at those isolated points, which are
polygonal corners of the second category. But if the group belong to the
set which includes the remaining families, then R0 and R0' are contiguous
along all edges of the second kind, and they may be contiguous also at
isolated points as before.

In the former case R0 and R0' may be regarded as distinct spaces,
each fundamental for its own half-plane. Let RQ have 2n edges which can
be arranged in n conjugate pairs, and let q be the number of cycles all
of which are closed ; each point in one edge corresponds to a single point in
the conjugate edge. Let the surface included by the polygon R0 be deformed
and stretched in such a manner that conjugate edges are made to coincide by
the coincidence of corresponding points. A closed surface is obtained. For
each pair of edges in the polygon there is a line on the surface, and for each
cycle in the polygon there is a point on the surface in which lines meet ; and
the lines make up a single curvilinear polygon occupying the whole surface.
The process is reversible ; and therefore the connectivity of the surface is an
integer which may properly be associated with the fundamental polygon.

When two consecutive edges are conjugate, their common corner is a
cycle by itself. The line, made up of these two edges after the deformation,
ends in the common corner which has become an isolated point ; this line
can be obliterated without changing the connectivity. The obliteration
annuls two edges and one cycle of the original polygon : that is, it diminishes
n by unity and q by unity. Let there be r such pairs of consecutive edges.
The deformed surface is now occupied by a single polygon, with n — r sides
and q — r angular points; so that, if its connectivity be 2N+1, we have

(§ 165)

2N= 2 + (n - r) - 1 - (q - r)

= n + l-q.
The group may be said to be of class N.

293.] CLASS OF GROUPS 609

In the latter case, the combination of R0 and R0' may be regarded as
a single region, fundamental for the whole plane. Let R0 have 2n edges of
the first kind and ra of the second kind, and let q be the number of closed
cycles : the number of open cycles is m. Then R0' has 2n edges of the first
kind and q closed cycles; it has, in common with JR0, the m edges of the
second kind and the m open cycles. The correspondence of points on the
edges of the first kind is as before. Let the surface included by R0 and
R0' taken together be deformed and stretched in such a manner that con
jugate edges coincide by the coincidence of corresponding points on those
edges. A closed surface is obtained. As the process is reversible, the
connectivity of the surface thus obtained is an integer which may properly
be associated with the fundamental polygon.

This integer is determined as before. For each pair of edges of the first
kind in either polygon, a line is obtained on the surface ; so that Zn lines are
thus obtained, n from _R0 and n from R0'. Each of the common edges of the
second kind is a line on the surface, so that m lines are thus obtained. The
total number of lines is therefore 2n + m. For each of the closed cycles
there is a point on the surface in which lines, obtained through the defor
mation of edges of the first kind, meet: their number is 2q, each of the
polygons providing q of them. For each of the open cycles there is a point
on the surface in which one of the m lines divides one of the n lines arising
through R0 from the corresponding line arising through -R0' : the number of
these points is m. The total number of points is therefore 2q + m.

The total number of polygons on the surface is 2. Hence, if the con
nectivity be 2N + 1, we have (§ 165)

2N = 2 + 2n + m - (2q + m) - 2

= 2w - 2q.
The group may be said to be of class N.

Thus for the generating quadrilateral in figure 112 (p. 596), the class of
the group is zero when the arrangement of the conjugate pairs is 1, 2 ; 3, 4 :
and it is unity when the arrangement of the pairs is 1, 3 ; 2, 4. For the
generating hexagon in figure 113 (p. 597), the class of the group is zero when
the arrangement of the conjugate pairs is 1, 6 ; 2, 5 ; 3, 4 : and it is unity
when the arrangement of the pairs is 1, 4; 2, 5 ; 3, 6. For the generating
pentagon in figure 114 (p. 597), the class of the group is zero when the
arrangement of the conjugate pairs is 1, 3; 4, 5 : arid it is two, when the
arrangement of the pairs is 1, 4; 3, 5. For a generating polygon, bounded
by 2n semi-circles each without all the others and by the portions of the
axis of x, the number of closed cycles is zero : hence N=n.

294. In all the groups, which lead to a division of a half-plane or of a
F. 39

610 KLEINIAN [294.

circle into polygons, the substitutions have real coefficients or are composed
of real substitutions and a single substitution with complex coefficients:
and thus the variation in the complex part of the coefficients in the group is
strictly limited. We now proceed to consider groups of substitutions

in which the coefficients are complex in the most general manner: such
groups, when properly discontinuous, are called Kleinian, by Poincare.

The Fuchsian groups conserve a line, the axis of x, or a circle, the funda
mental circle : the Kleinian groups do not conserve such a line or circle,
common to the group. Every substitution can be resolved into two dis
placements of origin, an inversion and a reflexion, as in § 258. The inversion
has for its centre the point — 8/7, being the origin after the first displace
ment ; the reflexion is in the line through this point making with the real
axis an angle TT - 2 arg. 7. The only line left unaltered by these processes is
one which makes an angle £TT - arg. 7 with the real axis and passes through
the point ; and the final displacement to the point 0/7 will in general displace
this line. Moreover, arg. 7 is not the same for all substitutions; there is
therefore no straight line thus conserved common to the group.

Similar considerations shew that there is no fundamental circle for the
group, persisting untransformed through all the substitutions.

Hence the Kleinian groups conserve no fundamental line and no funda
mental circle : when they are used to divide the plane, the result cannot be
similar to that secured by the Fuchsian groups. As will now be proved,
they can be used to give relations between positions in space, as well as
relations between positions merely in a plane.

The lineo-linear relation between two complex variables, expressed as a
linear substitution, has been proved (§ 261) to be the algebraical equivalent
of any even number of inversions with regard to circles in the plane of the
variables : this analytical relation, when developed in its geometrical aspect,
can be made subservient to the correlation of points in space.

Let spheres be constructed which have, as their equatorial circles, the
circles in the system of inversions just indicated; let inversions be now carried
out with regard to these spheres, instead of merely with regard to their
equatorial circles. It is evident that the consequent relations between points
in the plane of the variable z are the same as when inversion is carried out
with regard to the circles : but now there is a unique transformation of points
that do not lie in the plane. Moreover, the transformation possesses the
character of conformal representation, for it conserves angles and it secures
the similarity of infinitesimal figures: points lying above the plane of z

294.] GROUPS 611

invert into points lying above the plane of z, so that the plane of z is
common to all these spherical inversions and therefore common to the sub
stitutions, the analytical expression of which is to be associated with the
geometrical operation ; and a sphere, having its centre in the plane of the
complex z is transformed into another sphere, having its centre in that plane,
so that the equatorial circles correspond to one another.

Through any point P in space, let an arbitrary sphere be drawn, having
its centre in the plane of the complex variable, say, that of the coordinates
£, r). It will be transformed, by the various inversions indicated, into another
sphere, having its centre also in the plane of £, ij and passing through the
point Q obtained from P as the result of all the inversions; and the equatorial
planes will correspond to one another.

Let the sphere through Q be

Hence, if Q be determined by

z'=?+ irj', zj = r - vn1, p2 = p + v2 + r = *v + r/2,

this equation is p/2 + h0z' + hz0' + k = 0,

where — h, — h0 = a+ib, a— ib respectively. The equatorial circle of this
sphere is evidently given by £ = 0, so that its equation is

Z'ZQ + h^z' + hz0' + k = Q;

this circle can be obtained from the equatorial circle of the sphere through P
by the substitution z' = — — ~ . Hence the latter circle, by § 258, is given by

zz<> (a«0 + A0«7o + haQy + kyy0) + z0 («0

+ z (a/30 + h0aS0 + A/30y + &y80) + j3/30 + A0/3S0 + h/308 + k880 = 0 :
and therefore the equation of the sphere through P is

p2 (a«0 + A0«7o + ha0y + ky<y0) + z0 (a0/3 + h0@y0 + haa8 + ky08)

+ z (a/30 + h0«SQ + h&y + fcy80) + j3/30 + h0j3S0 + h/30S + kSS, = 0.

The quantities h, h0, k are arbitrary quantities, subject to only the single
condition that the sphere passes through the point Q: there is no other
relation that connects them. Hence the equation of the sphere through P
must, as a condition attaching to the quantities h, h0, k, be substantially the
equivalent of the former condition given by the equation of the sphere
through Q. In order that these two equations may be the same for h, hn, k,
the variables p'2, z', z0' of the point Q and those of P, being p2, z, z0, must give

39—2

612 CORRESPONDENCE OF POINTS IN SPACE [294.

practically the same coefficients of h, h0, k in the two equations, and therefore

= z :

These are evidently the equations which express the variables of a point Q in
space in terms of the variables of the point P, when it is derived from P by
the generalisation of the linear substitution

aw + ft

yw + 8'

they may be called the equations of the substitution. It is easy to deduce
that

which may be combined with the preceding equations of the substitution.

Also, the magnification for a single inversion is dsjds, or rjr, where rx
and r are the distances of the arcs from the centre of the sphere relative to
which the inversion is effected. But n/r = £/£ where £i and £ are the
heights of the arcs above the equatorial plane ; hence the magnification is
£"!/£, for a single inversion. For the next inversion it is fa/^i, and therefore it
is £2/£ for the two together; and so on. Hence the final magnification m
for the whole transformation is

m = -77 =

a quantity that diminishes as the region recedes from the equatorial plane.

It is justifiable to regard the equations obtained as merely the generalisa
tion of the substitution : they actually include the substitution in its original
application to plane variables. When the variables are restricted to the plane
of %, 77, we have p2 = zz0, and therefore

z = —

880 yz + 8 '

on the removal of the factor <y0z0 + 80 common to the numerator and the
denominator ; and £' vanishes when £ = 0. The uniqueness of the result is
an a posteriori justification of the initial assumption that one and the same
point Q is derived from P, whatever be the inversions that are equivalent to
the linear substitution.

294.] KLEINIAN GROUPS 613

Ex. 1. Let an elliptic substitution have u and v as its fixed points.

Draw two circles in the plane, passing through u and v and intersecting at an angle
equal to half the argument of the multiplier. The transformation of the plane, caused by
the substitution, is equivalent to inversions at these circles ; the corresponding transforma
tion of the space above the plane is equivalent to inversions at the spheres, having these
circles as equatorial circles. It therefore follows that every point on the line of intersection
of the spheres remains unchanged : hence when a Kleinian substitution is elliptic, every
point on the circle, in a plane perpendicular to the plane of x, y and having the line joining
the common points of the substitution as its diameter, is unchanged by the substitution.
Poincare calls this circle C the double, (or fixed) circle of the elliptic substitution.

Ex. 2. Prove that, when a Kleinian substitution is hyperbolic, the only points in space,
which are unchanged by it, are its double points in the plane of x, y ; and shew that
it changes any circle through those points into itself and also any sphere through those
points into itself.

Ex. 3. Prove that, when the substitution is loxodromic, the circle C, in a plane
perpendicular to the plane x, y and having as its diameter the line joining the common
points of the substitution, is transformed into itself, but that the only points on the
circumference left unchanged are the common points.

Ex. 4. Obtain the corresponding properties of the substitution when it is parabolic.
(All these results are due to Poincare".)

295. The process of obtaining the division of the ^-plane by means of
Kleinian groups is similar to that adopted for Fuchsian groups, except
that now there is no axis of real quantities or no fundamental circle
conserved in that plane during the substitutions : and thus the whole
plane is distributed. The polygons will be bounded by arcs of circles as
before : but a polygon will not necessarily be simply connected. Multiple
connectivity has already arisen in connection with real groups of the third
family by taking the plane on both sides of the axis.

As there are no edges of the second kind for polygons determined by
Kleinian groups, the only cycles of corners of polygons are closed cycles ;
let AQ, A1} ..., An-! in order be such a cycle in a polygon R0. Round AQ
describe a small curve, and let the successive polygons along this curve be
.Ro, -fin •••> -fin-i, -fin. — The corner A0 belongs to each of these polygons:
when considered as belonging to Rm, it will in that polygon be the homologue
of Am as belonging to R0, if m<n\ but, as belonging to Rn, it will, in that
polygon, be the homologue of A0 as belonging to fi0. Hence the substitution,
which changes ,R0 into Rn, has A0 for a fixed point.

This substitution may be either elliptic or parabolic, (but not hyperbolic,
§ 292) : that it cannot be loxodromic may be seen as follows. Let pei<a be
the multiplier, where (§ 259) p is not unity and to is not zero : and let
S0 denote the aggregate of polygons R0, Rlt ..., Rn-i, Si the aggregate
Rn, ..., -R2n_i, and so on. Then S0 is changed to S1( Si to 22, and so on;
by the substitution. Let p be an integer such that pa> ^ 2?r ; then, when

614 DIVISION OF SPACE [295.

the substitution has been applied p times, the aggregate of the polygons
is Sp, and it will cover the whole or part of one of the aggregates 20, 2i,«...
But, because p? is not unity, Sp does not coincide with that aggregate or the
part of that aggregate : the substitution is not then properly discontinuous,
contrary to the definition of the group. Hence there is no loxodroraic
substitution in the group. If the substitution be elliptic, the sum of the
angles of the cycle must be a submultiple of 2vr ; when it is parabolic, each
angle of the cycle is zero.

In the generalised equations whereby points of space are transformed
into one another, the plane of x, y is conserved throughout : it is
natural therefore to consider the division of space on the positive side of
this plane into regions P0, Pa,..., such that P0 is changed into all the
other regions in turn by the application to it of the generalised equations.
The following results can be obtained by considerations similar to those
before adduced in the division of a plane*.

The boundaries of regions are either portions of spheres, having their
centres in the plane of x, y, or they are portions of that plane : the
regions are called polyhedral, and such boundaries are called faces. If the
face is spherical, it is said to be of the first kind: if it is a portion of
the plane of x, y, it is said to be of the second kind. Faces of the
second kind, being in the plane of x, y and transformed into one another,
are polygons bounded by arcs of circles.

The intersections of faces are edges. Again, an edge is of the first
kind, when it is the intersection of two faces of the first kind : it is of i
the second kind, when it is the intersection of a face of the first kind
with one of the second kind. An edge of the second kind is a circular
arc in the plane of x, y: an edge of the first kind, being the intersection
of two spheres with their centres in the plane of x, y, is a circular arc,
which lies in a plane perpendicular to the plane of x, y and has its
centre in that plane.

The extremities of the edges are corners of the polyhedra. They are
of three categories :

(i) those which are above the plane of x, y and are the common
extremities of at least three edges of the first kind:

(ii) those which lie in the plane of x, y and are the common extremities
of at least three edges of the first kind :

(iii) those which lie in the plane of x, y and are the common extremities
of at least one edge of the first kind and of at least two edges of
the second kind.

* See, in particular, Poincar6, Acta Math., t. iii, pp. fifi et seq.

295.] FUNDAMENTAL POLYHEDEA AND POLYGONS 615

Moreover, points at which two faces touch can be regarded as isolated corners,
the edges of which they are the intersections not being in evidence.

Faces of a polyhedron, which are of the first kind, are conjugate in pairs :
two conjugate faces are congruent by a fundamental substitution of the group.

Edges of the first kind, being the limits of the faces, arrange themselves
in cycles, in the same way as the angles of a polygon in the division of the
plane. If E0, E1} ..., En^ be the n edges in a cycle, the number of regions
which have an edge in E0 is a multiple of n : and the sum of the dihedral
angles at the edges in a cycle (the dihedral angle at an edge being the
constant angle between the faces, which intersect along the edge) is a
submultiple of 2?r.

The relation between the polyhedral divisions of space and the polygonal
divisions of the plane is as follows. Let the group be such as to cause the
fundamental polyhedron P0 to possess n faces of the second kind, say F01,
FW, ..., Fm. Every congruent polyhedron will then have n faces of the
second kind; let those of Pg be Fsl, FS2, ..., Fsn. Every point in the plane
of x, y belongs to some one of the complete set of faces of the second kind :
and, except for certain singular points and certain singular lines, no point
belongs to more than one face, for the proper discontinuity of the group
requires that no point of space belongs to more than one polyhedron.

Then the plane of x, y is divided into n regions, say D:, D2, ..., Dn; each
of these regions is composed of an infinite number of polygons, consisting of
the polygonal faces F. Thus Dr is composed of Fw, Flr, F2r, ... ; and these
polygonal areas are such that the substitution Ss transforms Fw into F^.
Hence it appears that, by a Kleinian group, the whole plane is divided into
a finite number of regions ; and that each region is divided into an infinite
number of polygons, which are congruent to one another by the substitutions
of the group.

296. The preceding groups of substitutions, that have complex co
efficients, have been assumed to be properly discontinuous.

Ex. Prove that, if any group of substitutions with complex coefficients be improperly
discontinuous, it is improperly discontinuous only for points in the plane of x, y.

(Poincare.)

One of the simplest and most important of the improperly discontinuous
groups of substitutions, is that compounded from the three fundamental
substitutions

z' = Sz = z + I, z'=Tz = -- z' = Vz = z + i,

z

where i has the ordinary meaning. All the substitutions are easily proved to
be of the form

616 EXAMPLE OF AN IMPROPERLY [296.

where a.8 — 0y = I, and a, /3, 7, S are complex integers, that is, are represented
by m + ni, where m and n are integers. This is the evident generalisation of
the modular-function group: consequently there is at once a suggested
generalisation to a polyhedron of reference, bounded by

i^l^-i, i-^T^-i, £2 + r-K2^l,
which will thus have one spherical and four (accidentally) plane faces.

The following method of consideration of the points included by the
polyhedron of reference differs from that which was adopted for the polygon
of reference in the plane.

If possible, let a point (£, 77, £) lying within the above region be transformed
by the equations generalised from some one substitution of the group, say

from - — ~ , into another point of the region, say £', 77', £". Then we have

*>*>-*, i>i>-i. r+^'+r>i.

From the last, it follows that £ > -^ : and similarly for £', 77', £', by the

V*
hypothesis that the point is in the region. Now

1 1

and therefore !/(££') = 7 2 + ~ \ jz + $ |2.

Hence, as £ and £' are both >-r^, we have |7|2<2: so that, because 7 is

V^
a complex integer, we have

7 = 0, ±1, ±i
as the only possible cases.

If 7 = 0, then since «S — fty = 1, we have «S = 1 and a, S are complex
integers : thus either

a = 1 a. = — 1 a = i a = — i

f , or > , or

b = — 1 j b = — i) o =

For the first of these sub-cases we have, from the equations of the substitu
tion,

where /3 is a complex integer : if the new point lie within the region, then
/3 = 0, and we have

/ G»/ c»

z = z, £ = £
which is merely an identity.

For the second, we have / = z — ft : leading to the same result.
For the third, we have, since 80 = i,

296.] DISCONTINUOUS GROUP 617

But as ||'|, I?/ , |£|, 77] are all less than \, we have /3 = 0, and so

For the fourth case, we have

z' = -z- i/3,

leading to the same result as the third. Hence, if 7 — 0, the only point lying
within the region is given by

r = -£ */=-<?, r/ = ?=

ti/j
determined by the substitution w' = — ., which is TVT-1V~1TV.

v

If 7 = 1, that is, 770 = 1, then

Of the two quantities £ and f ', one will be not greater than the other : we
choose £ to be that one and consider the accordingly associated substitution * :
thus £/£'<!, p*>l, and so

20j0B + zyS0 + 880 < 0,

say *0? + 3+^«>.

7 7o 77o

Now (7] = 1, so that - is of the form p + iq, where p and q are integers : thus
we have

p2 + qz + 2p% + 2q<q < 0,
which is impossible because 2£ < 1, 2rj < 1.

Hence it follows that within the region there are only two equivalent
points, derived by the generalised equations from the substitution

, iw
»=-.;

and that all points within the region can be arranged in equivalent pairs

£ 77, f and - |, - 77, £

If the region be symmetrically divided into two, so that the boundaries of
a new region are

then no point within the new region is equivalent to any other point in the
regionf. As in the division of the plane by the modular group, it is easy
to see that the whole space above the plane of |, 77 is divided by the group :
therefore the region is a polyhedron of reference for the group composed of the
fundamental substitutions S, T, V.

* Were it f ', all that would be necessary would be to take the inverse substitution,
t Bianchi, Math. Ann., t. xxxviii, (1891), pp. 313—324, t. xl, (1892), pp. 332—412; Picard, ib.,
t. xxxix, (1891), pp. 142—144; Mathews, Quart. Journ. Math., vol. xxv, (1891), pp. 289—296.

618 EXAMPLE [296.

The preceding substitutions, with complex integers for coefficients, are of use in appli
cations to the discussion of binary quadratic forms in the theory of numbers. The special
division of all space corresponds, of course, to the character of the coefficients in the
substitutions : other divisions for similar groups are possible, as is proved in Poincare's
memoir already quoted.

These divisions all presuppose that the group is infinite : but similar divisions for only
finite groups (and therefore with only a finite number of regions) are possible. These are
considered in detail in an interesting memoir by Goursat* ; the transformations conserve
an imaginary sphere instead of a real plane as in Poincare's theory.

Ex. Shew that, for the infinite group composed of the fundamental substitutions

where e is a primitive cube root of unity, a fundamental region for the division of space
above the plane of z, corresponding to the generalised equations of the group, is a sym
metrical third of the polyhedron extending to infinity above the sphere

and bounded by the sphere and the six planes

2f= ±1, f + W3=±l, f-W3=±l. (Bianchi.)

* " Sur les substitutions orthogonales et les divisions regulieres de 1'espace," Ann. de VEc.
Norm. Sup., 3me Ser., t. vi, (1889), pp. 9—102. See also Schonflies, Math. Ann., t. xxxiv, (1889),
pp. 172 — 203 : other references are given in these papers.

CHAPTER XXII.

AUTOMORPHIC FUNCTIONS.

297. As was stated in the course of the preceding chapter, we are
seeking the most general form of the arguments of functions which secures
the property of periodicity. The transformation of the arguments of trigo
nometrical and of elliptic functions, which secures this property, is merely a
special case of a linear substitution : and thus the automorphic functions to
be discussed are such as identically satisfy the equation

F(SiZ)=f(z},

where Si is any one of an assigned group of linear substitutions of which only
a finite number are fundamental.

Various references to authorities will be given in the present chapter, in connection
with illustrative examples of automorphic functions : but it is, of course, beyond the scope
of the present treatise, dealing only with the generalities of the theory of functions, to
enter into any detailed development of the properties of special classes of automorphic
functions such as, for instance, those commonly called polyhedral and those commonly
called elliptic-modular. Automorphic functions, of types less special than those just men
tioned, are called Fuchsian functions by Poincare, when they are determined in association
with a Fuchsian group of substitutions, and Kleinian functions, when they are determined
in association with a Kleinian group : as our purpose is to provide only an introduction
to the theory, the more general term automorphic will be adopted.

The establishment of the general classes of automorphic functions is effected by
Poincare" in his memoirs in the early volumes of the Acta Mathematica, and by Klein in his
memoir in the 21st volume of the Mathematische Annalen : these have been already quoted
(p. 583 note) : and Pomcare" gives various historical notes* on the earlier scattered occur
rences of automorphic functions and discontinuous groups. Other memoirs that may be
consulted with advantage are those of Von Mangoldtf, Weber J, Schottky§, Stahl|!,

* Acta Math., t. i, pp. 61, 62, 293 : ib., t. iii, p. 92. Poincare's memoirs occur in the first,
third, fourth and fifth volumes of this journal : a great part of the later memoirs is devoted to
their application to linear differential equations.

t Gott. Nachr., (1885), pp. 313— 31(J ; ib., (1886), pp. 1—29.

J Gott. Nachr., (1886), pp. 359—370.

§ Crelle, t. ci, (1887), pp. 227—272.

|| Math. Ann., t. xxxiii, (1889), pp. 291—309.

ANHARMONIC GROUP AND FUNCTION [297.

Schlesinger* and Rittert : and there are two by BurnsideJ, of special interest and
importance in connection with the third of the seven families of groups (§ 292).

298. We shall first consider functions associated with finite discrete
groups of linear substitutions.

There is a group of six substitutions

1 1 z-l z

z, - , 1 — *,

s' ' \-z' z ' z-l'

which (§ 283) is complete. Forming expressions z— x, z -- , z — (\—x),

CC

T* *•-" I or

z — — =- and multiplying them together, we can express

CC .I

CO

their product in the form

so that

is a function of z which is unaltered by any of the transformations of its
variable given by the six substitutions of the group. The function is well
known, being connected with the six anharmonic ratios of four points in a
line which can all be expressed in terms of any one of them by means of the
substitutions.

Another illustration of a finite discrete group has already been furnished
in the periodic elliptic transformation of § 258, whereby a crescent of
the plane with its angle a submultiple of 2?r was successively transformed,
ultimately returning to itself: so that the whole plane is divided into portions
equal in number to the periodic order of the substitution.

If a stereographic projection of the plane be made with regard to any
external point, we shall have the whole sphere divided into a number of
triangles, each bounded by two small circles and cutting at the same angle.
By choice of centre of projection, the common corners of the crescents can be
projected into the extremities of a diameter of the sphere : and then each of
the crescents is projected into a lune. The effect of a substitution on the
crescent is changed into a rotation round the diameter joining the vertices
of a lune through an angle equal to the angle of the lune.

299. This is merely one particular illustration of a general correspondence
between spherical rotations and plane homographies, as we now proceed to
shew. The general correspondence is based upon the following proposition
due to Cay ley: —

* Crelle, t. cv, (1889), pp. 181—232.

t Math. Ann., t. xli, (1892), pp. 1—82.

J Land. Math. Soc. Proc., vol. xxiii, (1892), pp. 48—88, ib., pp. 281—295.

299.] HOMOGRAPHY AND ROTATIONS 621

When a sphere is displaced by a rotation round a diameter, the variables of
the stereographic projections of any point in its original position and in its dis
placed position are connected by the relation

, _(d + ic) z — (b — ia)
~ (b + ia) z + (d- ic) '

where a, b, c, d are real quantities.

Rotation about a given diameter through an assigned angle gives a
unique position for the displaced point : and stereographic projection, which
is a conformal operation in that it preserves angles, also gives a unique point
as the projection of a given point. Hence taking the stereographic projec
tion on a plane of the original position and the displaced position of a point
on the sphere, they will be uniquely related : that is, their complex variables
are connected by a lineo-linear relation, which thus leads to a linear substitu
tion for the plane-transformation corresponding to the spherical rotation.

Now the extremities of the axis are unaltered by the rotation ; hence the
projections of these points are the fixed points of the substitution. If the
points be £ 77, £ and - £, — 77, - £, on a sphere of radius unity, and if the
origin of projection be the north pole of the sphere, the fixed points of the
substitution are

_

so that the substitution is of the form

j % + w f+5"
i-r ~i^y

To determine the multiplier K, we take a point P very near C, one extremity
of the axis : let P' be the position after the rotation, so that GP' = GP. Then,
in the stereographic projection, the small arcs which correspond to GP and
GP' are equal in length, and they are inclined at an angle a. Hence the

multiplier K is eia : for when z, and therefore /, is nearly equal to — ~- — , a

fixed point of the substitution, the magnification is \K and the angular
displacement is the argument of K, which is a.

Inserting the value of K, solving for z' and using the condition
I2 + rf + £2 = 1> we have

,_(d + ic) z — (b — ia)
~ ~~

where a = 1~ sin £a, b = 77 sin £a, c = f sin £a, d = cos £a,

so that a2 + b2 + ca + d* = 1,

the equivalent of the usual condition to which the four coefficients in any

622 HOMOGENEOUS SUBSTITUTIONS [299.

linear substitution are subject : it is evident that the substitution is elliptic.
The proposition* is thus proved.

When the axis of rotation is the diameter perpendicular to the plane, we
have, by § 256,

z = ke~*+i*> z' = ke-*+i^+a\
so that z' = zeia,

agreeing with the above result by taking £ = 0 = 77, £=1 so that a = 0 = 6,
c = sin |a, d = cos |a.

It should be noted that the formula gives two different sets of coefficients
for a single rotation : for the effect of the rotation is unaltered when it is
increased by 2?r, a change in a which leads to the other signs for all the
constants a, b, c, d.

It thus appears that the rotation of a sphere about a diameter interchanges
pairs of points on the surface, the stereographic projections of which on the
plane of the equator are connected by an elliptic linear substitution : hence,
in the one case as in the other, the substitution is periodic when a, the
argument of the multiplier and the angle of rotation, is a submultiple of 2?r.

In the discussion of functions related in their arguments to these linear
substitutions, it proves to be convenient to deal with homogeneous variables,
so that the algebraical forms which arise can be connected with the theory of
invariants. We take zz2 = zl : the formulae of transformation may then be
represented by the equations

Zl = K (dZi + 0Za\ Za = K (jZi + BZ2)

for the substitution z' = (ctz + f3}/(yz + 8). As we are about to deal with
invariantive functions of position dependent upon rotations, it is important
to have the determinant of homogeneous transformation equal to unity.
This can be secured only if K — + 1 or if K = - 1 : the two values correspond
to the two sets of coefficients obtained in connection with the rotation.
Hence, in the present case, the formulae of homogeneous transformation are

z\ = (d + ic) Zi-(b- id) za, za' = (64- id) z^+(d- ic] za,

where o? + 62 + c2 4- d2, being the determinant of the substitution, = 1 ; every
rotation leads to two pairs of these homogeneous equations "f*. Each pair of
equations will be regarded as giving a homogeneous substitution.

Moreover, rotations can be compounded : and this composition is, in the
analytical expression of stereographically projected points, subject to the same
algebraical laws as is the composition of linear substitutions. If, then, there

* Cayley, Math. Ann., t. xv, (1879), pp. 238 — 240; Klein's Vorlesungen iiber das Ikosaeder,
pp. 32—34.

t The succeeding account of the polyhedral functions are based on Klein's investigations,
which are collected in the first section of his Vorlesungen iiber das Ikosaeder (Leipzig, Teubner,
1884) : see also Cayley, Camb. Phil. Trans., vol. xiii, pp. 4 — 68.

It will be seen that the results are intimately related to the results obtained in §§ 271 — 279,
relative to the conformal representation of figures, bounded by circular arcs, on a half-plane.

299.] GROUPS FOR THE REGULAR SOLIDS 623

be a complete group of rotations, that is, a group such that the composition
of any two rotations (including repetitions) leads to a rotation included in the
group, then there will be associated with it a complete group of linear
homogeneous substitutions. The groups are finite together, the number of
members in the group of homogeneous substitutions being double of the
number in the group of rotations : and the substitutions can be arranged in
pairs so that each pair is associated with one rotation.

300. Such groups of rotations arise in connection with the regular solids.
Let the sphere, which circumscribes such a solid, be of radius unity : and let
the edges of the solid be projected from the centre of the sphere into arcs of
great circles on the surface. Then the faces of the polyhedron will be repre
sented on the surface of the sphere by closed curvilinear figures, the angular
points of which are summits of the polyhedron. There are rotations, of proper
magnitude, about diameters properly chosen, which displace the polyhedron
into coincidence (but not identity) with itself, and so reproduce the above-
mentioned division of the surface of the sphere : when all such rotations have
been determined, they form a group which may be called the group of the
solid. Each such rotation gives rise to two homogeneous substitutions, so
that there will thence be derived a finite group of discrete substitutions:
and as these are connected with the stereographic projection of the sphere,
they are evidently the group of substitutions which transform into one
another the divisions of the plane obtained by taking the stereographic
projection of the corresponding division of the surface of the sphere. For
the construction of such groups of substitutions, it will therefore be sufficient
to obtain the groups of rotations, considered in reference to the surface of
the sphere.

I. The Dihedral Group. The simplest case is that in which the solid,
hardly a proper solid, is composed of a couple of coincident regular polygons
of n sides* : a reference has already been made to this case. We suppose the
polygons to lie in the equator, so that their corners divide the equator into
n equal parts : one polygon becomes the upper half of the spherical surface,
the other the lower half. The two poles of the equator, and the middle
points of the n arcs of the equator, are the corners of the corresponding solid.
Then the axes, rotations about which can bring the surface into such
coincidence with itself that its partition of the spherical surface is topo
graphically the same in the new position as in the old, are
(i) the polar axis,

(ii) a diameter through each summit on the equator,
(iii) a diameter through each middle point of an edge :
the last two are the same or are different according as n is odd or is even.
* The solid may also be regarded as a double pyramid.

1

624 DIHEDRAL GROUP AND FUNCTION [300.

For the polar axis, the necessary angle of rotation is an integral multiple

o__

of — . Thus we have £ = 0 =77, £= 1 and therefore
n

f\ 7 • T J "^

a = 0 = 6, c = sin - , a = cos - ;
n n

the substitutions are

imr _ ivr

Zi = 6 Z1} Z% = 6 Zz,

for r = 0, 1, .... w — 1. and

lirr inr

Zi= — en z1} z.2' = — e n z2,
for the same values of r. These are included in the set

jar iirr

z± = e zly z% =e z2,

for r = 0, 1, 2, ..., 2n — 1, being 2n in number: the identical substitution is
included for the same reason as before, when we associated a region of
reference in the ^-plane with the identical substitution.

For each of the axes lying in the equator, the angle of rotation is
evidently TT. Let an angular point of the polygon lie on the axis of £, say at
£ = 1; 77 = 0, £=0. Then so far as concerns (ii) in the above set, if we take

the axis through the (r + l)th angular point, we have £ = cos — — , 77 = sin -- ,

Tit H/

£=0 ; hence, as a is equal to TT, we have, for the corresponding substitutions,

Z±

for r = 0, 1, . . . , n — I. and

2nu ^Zmi

Z-^ — ^~ v(s ^2 > *^2 — ^ 1 9

for the same values of r.

And so far as concerns (iii) in the above set, if we take an axis through
the middle point of the rth side, that is, the side which joins the rth and the

(r + l)th points, then £ = cos ^ - -— , v =sin - - - — , £= 0 : hence as a

n n

is equal to TT, we have, for the corresponding substitutions,

^— —

Zi=e n z2,
for r = 0, 1, . . . , n — 1, and

Zi =-e n z.2) z« = — e n z1}
for the same values of r.

If n be even, the set of substitutions associated with (ii) are the same in
pairs, and likewise the set associated with (iii) ; if n be odd, the set associated
with (ii) is the same as the set associated with (iii). Thus in either case there
are 2n substitutions : and they are all included in the form

inr i^r

Zi = ie n £2, Zz = ie n z1}
for r=0, 1, ..., 2n-l.

300.] TETRAHEDRAL GROUP 625

Thus the whole group o/4n substitutions, in their homogeneous form, is

zl = en z1

iff

z'=e~^.

for r = 0, 1, ..., 2n - 1 : and in the non-homogeneous form, the group is

Ziwr

z =e

z, z' =

where r = 0, 1, ..., n-\ for each of them. The non- homogeneous expres
sions are not in their normal form in which the determinant of the coefficients
in the numerator and denominator is unity. Each expression gives two
homogeneous substitutions.

It is easy geometrically to see that all the axes have been retained : and
that they form a group, that is, composition of rotations about any two of the
axes is a rotation about one of the axes. The period for each of the equatorial

axes is 2 ; the period for a rotation - - about the polar axis depends on the
reducibility of - .

n

Before passing to the construction of the functions which are unaltered
for the dihedral group of substitutions, we shall obtain the tetrahedral group
and construct the tetrahedral functions, for the explanations in regard to the
dihedral functions arise more naturally in the less simple case.

II. The Tetrahedral Group. We take a regular cube as in the figure :
then ABCD is a tetrahedron, A'B'G'D' is the polar tetrahedron.

Fig. 119.
It is easy to see that the axes of rotation for the tetrahedron are

(i) the four diagonals of the cube A A', BB' , CC', DD' ;
F- 40

626 TETRAHEDRAL GROUP [300.

(ii) the three lines joining the middle points of the opposite edges of
the tetrahedron.

The latter pass through the centre of the cube and are perpendicular to
pairs of opposite faces. When the sphere circumscribing the cube is drawn,
the three axes in (ii) intersect the sphere in six points which are the angles
of a regular octahedron. Thus, though the axes of rotation for the three
solids are not the same, the tetrahedron, the cube, and the octahedron may
be considered together: in fact, in the present arrangement whereby the
surface of the sphere is considered, the cube is merely the combination of the
tetrahedron and its polar.

For each of the diagonals of the cube, the necessary angle of rotation
for the tetrahedron is 0 or |TT or |TT : the first of these gives identity, and
the others give two rotations for each of the four diagonals of the cube, so
that there are eight in all.

For each of the diagonals of the octahedron, the angle of rotation for
the tetrahedron is TT : there are thus three rotations.

With these we associate identity. Hence the number of rotations for the
tetrahedron is (8 + 3 + 1 =) 12 in all.

There are two sets of expressions for the tetrahedron according to the'
position of the coordinate axes of the sphere. One set arises when these are
taken along Ox, Oy, Oz, the diagonals of the octahedron ; the other arises,
when a coordinate plane is made to coincide with a plane of symmetry of the
tetrahedron such as B'DBD'.

Let the axes be the diagonals of the octahedron. The results are
obtainable just as before, and so may now merely be stated :

For OB', % = T] = % = -TK 5 when a = f TT, the substitution is

Y o

, z + i
z'=

z — i
and when a = |TT, the substitution is

Z =1

- r .

z — 1

For OA, £ = — 77 = f = - ; when a = |TT, the substitution is

yo

and when a. = |TT, the substitution is

- z ~

300.] OF SUBSTITUTIONS 627

For OC, - £ = 77 = £ = — ; when a = ITT, the substitution is

, .z-l
» — * ^ ,

and when a = |TT, the substitution is

z'=-

For OD', -£=-77 = £= — ; when a = f TT, the substitution i

, z — %

Z = ;

and when a = |TT, the substitution is

For Ox, t- = 1, ij = 0, f = 0 and a = TT : the substitution is

,'=i.

z
For #y, £ = 0, 17 = 1, £= 0, and a = vr : the substitution is

For Oz, % = 0, 77 = 0, f= 1 and a = TT : the substitution is

z = — z,
And identity is / = s.

Hence ^e group of tetrahedral non-homogeneous substitutions is

Ae a^es o/ reference in the sphere are the diameters bisecting opposite
edges of the tetrahedron. Each of these substitutions gives rise to two homo
geneous substitutions, making 24 in all.

To obtain the transformations in the case when the plane of xz is a plane
of symmetry of the tetrahedron passing through one edge and bisecting the
opposite edge, such as B'DBD' in the figure, it is sufficient to rotate the
preceding configuration through an angle \ir about the preceding 0^-axis,
and then to construct the corresponding changes in the preceding formulae.

For this rotation we have, with the preceding notation of § 299, £ = 0 = 17,
£= !> a=^'- then a = Q = b, c = sin ITT, d = cos^7r, so that d±ic = e±*"i:
and therefore the f of the displaced point in the stereographic projection is
connected with the f of the undisplaced point in the stereographic projection
by the equation

r%-lc

40—2

628 TETRAHEDRAL SUBSTITUTIONS [300.

If then Z be the variable of the projection of the undisplaced point and Z'
that of the projection of displaced point with the present axes, and z and z'
be the corresponding variables for the older axes, we have

„ l + i „, 1 + i ,

Z=^ZZ' Z-"^Z>

1 — i „ , \—ir7

that is, = ~V2~^' =~V2~

Taking now the twelve substitutions in the form of the last set and substi
tuting, we have a group of tetrahedral non-homogeneous substitutions in the
form

7'-, 7 ,1 ,
±t ±

when one of the coordinate planes is a plane through one edge of the
tetrahedron bisecting the opposite edge: each of these gives rise to two
homogeneous substitutions, making 24 in all.

301. The explanations, connected with these groups of substitutions,
implied that certain aggregates of points remain unchanged by the operations
corresponding to the substitutions. These aggregates are (i) the summits of
the tetrahedron, (ii) the summits of the polar tetrahedron — these two sets
together make up the summits of the cube: and (iii) the middle points of the
edges, being also the middle points of the edges of the polar tetrahedron —
this set forms the summits of an octahedron.

When these points are stereographically projected, we obtain aggregates
of points which are unchanged by the substitutions. We therefore project
stereographically with the extremity z of the axis Oz for origin of projection :
and then the projections of x, x', y, y' , z, z are 1, - 1, i, — i, <x> , 0, which are
the variables of these points.

Instead of taking factors z-l, z + l, ..., we shall take homogeneous
forms Z-L — ZZ, Zi + z2, z-^ — iz.^ zl + izz, z^, ^ ; the product of all these factors
equated to zero gives the six points. This product is

t = z& (zf - zf}.
For the tetrahedron ABCD, the summits A, B, C, D are -^ , -r* , -jn >i

1

73" -73 -73' 7S- 7s' v5' 73' and

therefore the variables of the points in the stereographic projection are

1 — i —l—i c n —l + i fn l+i

of A, -= -- ; of B, -7= - ; of C, -= -- ; of D, —j— - .
V3-1 V3 + 1 V3-1 V3+1

301.] TETRAHEDRAL FUNCTIONS 629

Forming homogeneous factors as before, the product of the four equated to
zero gives the stereographic projections of the four summits of the tetra
hedron A BCD. This product is

¥ = zj* - 2 x/^3*! V + *<?•

Similarly for the tetrahedron A'B'C'D'; the product of the factors
corresponding to the stereographic projections of its four summits is

3> = V + 2 V^3 *! V + ^24.

And the product of the eight points for the cube is <&W, that is,

W = zl*+ 14^ V + zf.

All these forms t, 4>, ^ are, by their mode of construction, unchanged
(except as to a constant factor, which is unity in the present case) by the
homogeneous substitutions : and therefore they are invariantive for the group
of 24 linear homogeneous substitutions, derived from the group of 12 non-
homogeneous tetrahedral substitutions. If W be taken as a binary quartic,
then <I> is its Hessian and t is its cubicovariant : the invariants are numerical
and not algebraical : and the syzygy which subsists among the system of
concomitants is

&-<¥* = 12 -\/^3P,

a relation easily obtained by reference merely to the expressions for the forms
<J>, ¥, t.

The object of this investigation is to form Z, the simplest rational
function of z which is unaltered by the group of substitutions: for this
purpose, it will evidently be necessary to form proper quotients of the
foregoing homogeneous forms, of zero dimensions in zl and #2. Let R
be any rational function of z, which is unaltered by the tetrahedral
substitutions. These substitutions give a series of values of z, for which
Z has only one value : hence R and Z, being both functions of z and
therefore of one another, are such that to a value of Z there is only one
value of R, so that .R is a rational function of Z.

In particular, the relation between R and Z may be lineo-linear : thus Z
is determinate except as to linear transformations. This unessential indeter-
minateness can be removed, by assigning three particular conditions to
determine the three constants of the linear transformation.

The number of substitutions in the z -group is 1 2 : hence as there will
thus be a group of 12 ^-points interchanged by the substitutions, the simplest
rational function of Z will be of the 12th degree in z, and therefore the
numerator and the denominator of the fraction for Z, in their homogeneous
forms, are of the 12th degree. The conditions assigned will be

(i) Z must vanish at the summits of the given tetrahedron :
(ii) Z must be infinite at the summits of the polar tetrahedron :
(iii) Z must be unity at the middle points of the sides.

630

TETRAHEDRAL

[301.

Then Z, being a fractional function with its numerator and its denominator
each of the 12th degree and composed of the functions <I>, ^ , t, must, with
the foregoing conditions, be given by

\I/3

by means of the syzygy, we have

Z : Z — 1 : 1 = "^ : — 12 \/^3£2 : <l>3,

which is Klein's result. Removing the homogeneous variables, we have
Z:Z-\:\=(&-2 V^3^2 + I)3 : - 12 \/^3^2 (z* - I)2 : (z* + 2 V^3^2 + I)3 ;

and then Z is a function of z which is unaltered by the group of 12 tetra-
hedral substitutions of p. 627. And every such function is a rational function
of Z.

This is one form of the result, depending upon the first position of the
axes : for the alternate form it is necessary merely to turn the axes through
an angle of ^TT round the ^-axis, as was done in § 300 to obtain the new
groups. The result is that a function Z, unaltered by the group of 12
substitutions of p. 628, is given by

Z : Z-l :l = (z*-2 \/3z2 - I)3 : - 12 \/3^2 <>4 + I)2 : (z* + 2 \/3> - I)3.

It still is of importance to mark out the partition of the plane corre
sponding to the groups, in the same manner as was done in the case of the
infinite groups in the preceding chapter. This partition of the plane is the
stereographic projection of the partition of the sphere, a partition effected by
the planes of symmetry of the tetrahedron. Some idea of the division may
be gathered from the accompanying figure, which is merely a projection on
the circumscribing sphere from the centre of the cube. The great circles

301.] FUNCTIONS 631

meet by threes in the summits of the tetrahedron and its polar, being the
sections by the three planes of symmetry, which pass through every such
summit, and the circles are equally inclined to one another there : they meet
by twos in the middle points of the edges and they are equally inclined to
one another there. They divide the sphere into 24 triangles, each of which
has for angles £TT, ^TT, £TT. (See Case II., § 278.)

The corresponding division of the plane is the stereographic projection of
this divided surface. Taking A as the pole of projection, which is projected

Fig. 121.

to infinity, then A' is the origin : the three great circles through A' become
three straight lines equally inclined to one another ; the other three great
circles become three circles with their centres on the three lines concurrent
in the origin. The accompanying figure shews the projection : the points in
the plane have the same letters as the points on the sphere of which they
are the projections : and the plane is thus divided into 24 parts. There are,
in explicit form, only 12 non-homogeneous substitutions: but each of these
has been proved to imply two homogeneous substitutions, so that we have
the division of the plane corresponding to the 24 substitutions in the group.
The fundamental polygon of reference is a triangle such as CA 'x'.

302. It now remains to construct the function for the dihedral group.
The sets of points to be considered are : —

(i) the angular points of the polygon : in the stereographic projection,
these are

Zirsi

e n , for s = 0, 1, ..., n— 1 ;

632 DIHEDRAL FUNCTION [302.

(ii) the middle points of the sides : in the stereographic projection,
these are

irt(2*+l)

e n , for s = 0, 1, . . ., n — I ; and
(iii) the poles of the equator which are unaltered by each of the

rotations : in the stereographic projection, these are 0 and oo .
Forming the homogeneous products, as for the tetrahedron, we have, for (i),

for (ii), V=z1n + z2n ;

and, for (iii), W = zlzz;

these functions being connected by a relation

_ [72 + y-2 _ 4 -yfn^

Because the dihedral group contains 2w non-homogeneous substitutions,
the rational function of z, say Z, must, in its initial fractional form, be of
degree 2n in both numerator and denominator ; and it must be constructed
from U, V, W.

The function Z becomes fully determinate, if we assign to it the following
conditions :

(i) Z must vanish at points corresponding to the summits of the
polygon,

(ii) Z must be infinite at points corresponding to the poles of the
equator,

(iii) Z must be unity at points corresponding to the middle points of

the edges :
and then we find

which gives the simplest rational function of z that is unaltered by the
substitutions of the dihedral group.

The discussion of the polyhedral functions will not be carried further here : sufficient
illustration has been provided as an introduction to the theory which, in its various
bearings, is expounded in Klein's suggestive treatise already quoted.

Ex. 1. Shew that the anharmonic group of § 298 is substantially the dihedral group
for n = 3 ; and, by changing the axes, complete the identification. (Klein.)

Ex. 2. An octahedron is referred to its diagonals as axes of reference, and a partition
of the surface of the sphere is made with reference to planes of symmetry and the axes of
rotations whereby the figure is made to coincide with itself.

Shew that the number of these rotations is 24, that the sphere is divided into 48
triangles, that the non-homogeneous substitutions which transform into one another the
partitions of the plane obtained from a stereographic projection are

n

(/

_ _ __ __ _

- 6-^) j v * (/ _ . fr .•1*' ^1

Z f— 1' 2 + 1' Z + l' Z-l'

where &=0, 1, 2, 3 ; and that the corresponding octahedral function is

Z.Z-l :l=(28 + 1424 + l)3:(2i2-3338-3324 + l)2:10824(24-l)*. (Klein.)

303.] ELLIPTIC MODULAR-FUNCTIONS 633

303. We now pass from groups that are finite in number to the
consideration of functions connected with groups that are infinite in
number. The best known illustration is that of the elliptic modular-
functions ; one example is the form of the modulus in an elliptic integral
as a function of the ratio of the periods of the integral. The general
definition of a modular -function* is that it is a uniform function such that

an algebraical equation subsists between ty (— — ^ j and ^r(w), where a, /3

7, 8 are integers subject to the relation aS — fiy = 1. The simplest case is
that in which the two functions i/r are equal.

The elliptic quarter-periods K and iK' are defined by the integrals

- to)} -* dz = [z (1 - *) (1 - cz)} -i dz,

Jo Jo

where c + c' = 1. The ordinary theory of elliptic functions gives the equation

dc dc 4cc' '

whatever be the value of c. To consider the nature of these quantities as
functions of c, we note that c = 1 is an infinity of K and an ordinary point of
K', and that c = 0 is a similar infinity of K' and an ordinary point of K : and
these are all the singular points in the finite part of the plane. The value
c = oo must also be considered. All other values of c are ordinary points for
K and K'.

For values of c, such that |c < 1, we have

so that, in the vicinity of the origin,

dc \KJ = ~ 4KW

1 (1 1 )

= K- + 9+ positive integral powers of c> .

7T 1C — I

Hence in the vicinity of the origin

K' 1

where P (c) is a uniform series converging for sufficiently small values of |c
and therefore, still in the vicinity of the origin,

Kf = - — \Qffc+KP(c).

7T

This is the definition of a modular-function which is adopted by Hermite, Dedekind, Klein,
Weber and others.

634 ELLIPTIC [303.

Now let the modulus c describe a contour round the origin and return to
its original value. Then K is unchanged, for the c-origin is not a singularity
ofK.

The new value of K' is evidently

that is, iK' changes into 2K + iK'. Hence, when c describes positively a
small contour round the origin, the quarter -periods K and iK' become K and
2K + iK' respectively.

In the same way from the equation

dK_ = _

dc' L dc''~ 4cc"

and from the expansion of K' in powers of c' when |c' < 1, we infer that
when c' describes positively a small contour round its origin, that is, when c
describes positively a small contour round the point c = 1, then iK' is unchanged
and K changes to K — 2iK'.

It thus appears that the quantities K and iK', regarded as functions of
the elliptic modulus c, are subject to the linear transformations

U(K) = K \ V(K) = K-2iK'l

U (iK') = 2K + iK'} ' V (iK') = iK'} '

without change of the quantity c ; and the application of either substitution
is equivalent to making c describe a closed circuit round one or other of the
critical points in the finite part of the plane, the description being positive if
the direct substitution be applied and negative if the inverse be applied.

When these substitutions are applied any number of times — the index
being the same and composed in the same way for K as for iK' — then,
denoting the composite substitution by P, we have results of the form

PiK' = /3K +

where ft and 7 are even integers, a and 8 are odd integers of the forms
1 + 4p, 1 + 4>q, say = 1 (mod. 4), and, because the determinant of U and that
of V are both unity, we have a.8 — (S<y = 1 by § 282. These equations give
the partially indeterminate form of the values of the quarter-periods for an
assigned value of the modulus c.

iK'

Conversely, we may regard c as a function of w = -=^- , the quotient of

the quarter-periods. The quotient is taken, for various reasons : thus it
enables us to remove common factors, it is the natural form in the passage
to g-series, and so on. The function is unaltered, when w is subjected

303.] MODULAR-FUNCTIONS 635

to the infinite group of substitutions derived from the fundamental
substitutions

.

1 — 2t0

Denoting the function c by </> (w), we have

We have still to take account of the relation of iK'jK to c, when the latter has
infinitely large values. For this purpose, we compare the differential expressions

which are equal to one another if kzx=y and H=l. As a? moves from 0 to 1, ?/ moves
from 0 to k'\ that is, from 0 to 1 /I'2 ; integrating between these limits, we have

where A and A' are quarter-periods with modulus l = \jk. As y moves from 0 to 1, x
moves from 0 to l/k'2 ; integrating between these limits, we have

so that kiK'=-iA'.

In order to obtain the effect on K and iK' of an infinitely large circuit described
positively by c, we make I describe a very small circuit round its origin negatively. By
what has been proved, the effect of the latter is to change A and iA' into A arid
iA' — 2A respectively. Hence the new value of TciK' is

- i\' + 2 A = k (3iK' + 2K);
and the new value of kK is

A + iA' - 2 A = - £ (2iA" + /iT).

Hence if w' denote the new value of w, consequent on the description of the infinitely
large circuit by c, we have

No new fundamental substitution is thus obtained ; and therefore U, V are the only
fundamental substitutions of the group for c, regarded as a modular- function.

Again, c' is a rational function of c and is therefore a modular-function :
consequently also cc' is a modular-function. Being a rational function of
c, it is subject to the two substitutions U and V, which are characteristically
fundamental for <f> (w). Now cc' is unchanged when we interchange c and c',
that is, when we interchange K and K' ; so that, if Kl and iKJ be new
quarter-periods for a modulus cc', we have

K, = K', iK,' = iK,

and therefore w,= -- .

to

Thus cc as a modular-function must be subject to the substitution

Tw = --.

w

636 MODULAR-FUNCTIONS [303.

But

- f ^- —- ,

UTw 2 + Tw 1 — 2w

so that V is compounded of T and U. Hence the substitutions for cc',
regarded as a modular-function, are the infinite group, derived from the
fundamental substitutions

Uw = w + 2, Tw = - - .

w

Denoting the modular-function cc' by % (w), we have

To obtain the change in w caused by changing c into c/c', we use the
differential expression

When the variable is transformed by the equation* (1 — y)(\ — k'2x) = l - x,
where k'-l2 = — k1, the expression becomes

When y describes the straight line from 0 to 1 continuously, x also
describes the straight line from 0 to 1 continuously. Integrating between
these limits, we have

A = k'K,

where A is a quarter-period. When y describes the straight line from 0
to 1/Z continuously, x describes the straight line from 0 to GO continuously
or, say, the line from 0 to 1/&2 and the line from i/k2 to oo continuously.
Integrating between these limits, we have

A + 1 A' = k'(K + iK') + \U \ {as (I- as) (1 -

J\

k"-

on using the transformation k2xu = 1 and taking account of the path described
by the variable u : and therefore

iA.' =k'(K + iK').

Hence the change of modulus from k to ik/k', which changes c to — c/c', gives
the changes of quarter-periods in the form

A = k'K, i\' = k'(K + iK');
and therefore the new value of w, say wl , is

wl = iv + 1 = Sw.

It therefore follows that, when c — c/c' is regarded as a modular function
of the quotient iv of the quarter-periods K and iK', it must be subject to
the substitutions

This is the equation expressing elliptic functions of k'u in terms of elliptic functions of u.

303.] AUTOMOKPHIC FUNCTIONS 637

Evidently S2 = U, and U may therefore be omitted ; V and S are the
fundamental substitutions of the infinite group of transformations of w,
the argument of the modular-function c2/c'.

As a last example, we consider the function

(c2-c + l)3

(c2-c)2

It is a rational function of cc', and therefore is a modular-function having the
substitutions Tw and Uw. By § 298, it is unaltered when we substitute

s\

— -- for c. It has just been proved that this change causes a change of w

C — -L

into w + 1, and therefore J, as a modular- function, must be subject to the
substitution

Sw = w + 1.

Evidently S2w — w + 2 = Uw, so that U is no longer a fundamental substitution
when S is retained. Hence we have the result that J is unaltered, when w is
subjected to the infinite group of substitutions derived from the fundamental
substitutions

so that we may write

Tw = -- ,

w

c- (c — I)2
This is the group of substitutions considered in § 284 : they are of the

form - — -r-, where a, /3, 7, 8 are real integers subject to the single relation
oto — /?7 = 1.

These illustrations, in connection with which the example in § 298 should be con
sulted, suffice to put in evidence the existence of modular-functions, that is, functions
periodic for infinite groups of linear substitutions, the coefficients of which are real
integers. The theory has been the subject of many investigations, both in connection
with the modular equations in the transformation of elliptic functions and also as a
definite set of functions. The investigations are due among others to Hermite, Fuchs,
Dedekind, Hurwitz and especially to Klein * ; and reference must be made to their
memoirs, or to Klein-Fricke's treatise on elliptic modular-functions, or to Weber's
Elliptische Functionen, for an exposition of the theory.

304. The method just adopted for infinite groups is very special, being
suited only to particular classes of functions : in passing now to linear
substitutions, no longer limited by the condition that their coefficients are
real integers, we shall adopt more general considerations. The chief
purpose of the investigation will be to obtain expressions of functions
characterised by the property of reproduction when their argument is
subjected to any one of the infinite group of substitutions.

* Some references are given in Enneper's Elliptische Functionen, (2tc Aufl.), p. 482.

638 CONSTRUCTION OF [304.

The infinite group is supposed of the nature of that in § 290 : the
members of it, being of the form

or

are such that a circle, called the fundamental circle, is unaltered by any of the
substitutions. This circle is supposed to have its centre at the origin and
unity for its radius.

The interior of the circle is divided into an infinite number of curvilinear
polygons, congruent by the substitutions of the group : each polygon contains
one, and only one, of the points in the interior associated by the substitutions
with a given point not on the boundary of the polygon. Hence corresponding
to any point within the circle, there is one and only one point within the
fundamental polygon, as there is only one such point in each of the polygons :
of these homologous points the one, which lies in the fundamental polygon
of reference, will be called the irreducible point. It is convenient to speak of
the zero of a function, implying thereby the irreducible zero : and similarly
for the singularities.

The part of the plane, exterior to the fundamental circle, is similarly
divided : and the division can be obtained from that of the internal area by
inversion with regard to the circumference and the centre of the fundamental
circle. Hence there will be two polygons of reference, one in the part of the
plane within the circle and the other in the part without the circle : and
all terms used for the one can evidently be used for the other. Thus the
irreducible homologue of a point without the circle is in the outer polygon
of reference : for a substitution transforms a point within an internal polygon
to a point within another internal polygon, and a point within an external
polygon to a point within another external polygon.

Take a point z in the interior of the circle, and round it describe a small
contour (say for convenience a circle) so as not to cross the boundary of the
polygon within which z lies : and let Zi be the point given by the substitution
fi(z). Then corresponding to this contour there is, in each of the internal
polygons a contour which does not cross the boundary of its polygon : and as
the first contour (say 00) does not occupy the whole of its polygon and as the
congruent contours do not intersect, the sum of the areas of all the contours
C{ is less than the sum of the areas of all the polygons, that is, the sum is
less than the area of the circle and so it is finite.

If ^ be the linear magnification at Zi, we have

1_ _ dzj,

\1iZ ~t~ Off dz
and therefore, if TO; be the least value of the magnification for points lying

within (70, we have

Ci > m-2C0.

304.]

A CONVERGING SERIES

639

*

The point is the homologue of z = oo by the substitution

/*

/ # .% _|_ P A

( z, - — — K* , and therefore — Si/yt lies without
\ 7i£ + OiJ

the circle : though, in the limit of i infinite, it
may approach indefinitely near to the circum
ference*.

Let this point be G: and through G and
0, the centre of the fundamental circle, draw
straight lines passing through the centre of
the circular contour. Then evidently

and, if Mt be the greatest magnification, then

1

GQ2'

so that

^

I ~ GQ- '

Fig. 122.

Now G is certainly not inside the circle, so that GQ is not less than RA :
thus

GQ

GQ

RAMA>

which is independent of the point G, that is, of the particular substitution

, ~D Z>\ 2

fi(z). Denoting f-gT-J by K, we have

or

Evidently //,; is finite.

Now

and therefore
so that

Mi<Kmi.

t=0

* For, in § 284, when the coefficients are real, a point associated with a given point may, for
i = co , approach indefinitely near to a point on the axis of x : and then, by the transformation of
§ 290, we have the result in the text.

640

A CONVERGING SERIES

[304.

It has been seen that S Ci is less than the area of the fundamental circle and

i = 0

is therefore finite : hence the quantity

i=0

is finite. It therefore follows that 2 p? is an absolutely converging series.

* = 0

00

Similarly, it follows that 2 /ifm is an absolutely converging series for all

i = 0

values of m that are greater than unity *. This series is evidently

i=0

and the absolute convergence is established on the assumption that z lies
within the fundamental circle.

Next, let z lie without the fundamental circle. If z coincide with some
one of the points — 81/7;, then the corresponding term of
the series

*=0

is infinite.

If it do not coincide with any one of the points
— Bi/ji , let c be its distance from the nearest of them, so

that

|7j£ + 8j ~2m < J7i ~2m c~2m.

Let z' be any point within the fundamental circle : then

Fig. 123.

, for any point within the circle, so that

Now Gz' < I + OG < 1 +

Hence

Only a limited number of the points - &{/% can be at infinity. Each of
the corresponding substitutions gives the point at infinity as the homologue
of — Si/ji ; and therefore, inverting with regard to the fundamental circle, we
have a number of homologues of the origin coinciding with the origin, equal
to the number of the points — 8^/7^ at infinity. The origin is not a singularity
of the group, so that the number of homologues of the origin, coincident with

it, must be limited.

•

* A completely general inference as to the convergence of the series, when m = l, cannot be
made : the convergence depends upon the form of the division of the plane into polygons, and
Burnside (I.e., p. 620) has proved that there is certainly one case in which S fr is an absolutely
converging series.

304.] CONNECTED WITH INFINITE GROUPS 641

Omitting the corresponding terms from the series, an omission which does
not affect its convergence, we can assign a superior limit to

C-l. Then

: let it be

fiS + Si-

2TO

mi /(l\2m °°

Ihus 2 7i2 + & ~2W < ( - 2 IW +

»=o W t-=0 ' X

which is a finite quantity by the preceding investigation, for z' is a point
within the circle.

Lastly, let z lie on the fundamental circle. If it coincide with one of the
essential singularities of the group, then there is an infinite number of points

- Si/vt which coincide with it : and so there will be an infinite number of
terms in the series infinite in value. If it do not coincide with any of the
essential singularities of the group, then there is a finite (it may be small,
but it is not infinitesimal) limit to its distance from the nearest of the points

- Bi/jt : the preceding analysis is applicable, and the series converges.

Hence, summing up our results, we have : —

00

The series 2 1 7^ + 8{ |~2m

z = 0

is an absolutely converging series for any point in the plane, which is not
coincident with any one of the points - B^ (which all lie without the funda
mental circle) or with any one of the essential singularities of the assigned
group (which all lie on the circumference of the fundamental circle)*.

305. Let H(z) denote a rational function of z, having a number of
accidental singularities a,,..., ap, no one of which lies on the fundamental
circle ; and let it have no other singularities. Consider the series

i=o

the group being the same as above. If z do not coincide with any of the
points au ..., ap, or with any of the points homologous with a1} ..., ap by the
substitutions of the group, there is a maximum value, say M, for the modulus

of H with any of the arguments °^ + ffi . Then

\®(z)\<M 1

The coefficients a, /3, j, S of the substitutions of the group depend upon the coefficients of
the fundamental substitutions, which may be regarded as parameters, arbitrary within limits.
The series is proved by Poincare to be a continuous function of these parameters, as well
as of the variable z : this proposition, however, belongs to the development of the theory and can
be omitted here as we do not propose to establish the general existence of all the functions.

F. 41

642 THETAFUCHSIAN FUNCTIONS [305.

and the right-hand side is finite, if in addition z do not coincide with any of
the points — 8^/7^ or with any of the essential singularities of the group.
Hence % (z) is an absolutely converging series for any value of z in the plane
which does not coincide with (i) an accidental singularity of H (z), or one of
the points homologous with these singularities by the substitutions of the
group, or with (ii) any of the points —Bifji, which are the various points
homologous with z = <x> by the substitutions of the group, or with (iii) any of
the essential singularities of the group, which are points lying on the funda
mental circle.

All these points are singularities of © (z\

If z coincide with/fc(a) and if fi{fk(z)} =z> then the term H\— -~ -^-M

is infinite, the point being an accidental singularity of H\— — **)• The

\tfi^ ' ®i'

rest of the series is then of the same nature as © (z) in the more general
case, and therefore converges. Hence the point is an accidental singularity
of the function © (z) of the same order as for H, that is, the series of points,
given by the accidental singularities of H (z) and by the points homologous
with them through the substitutions of the group, are accidental singularities
of the function © (z}.

In the same way it is easy to see that the points — Si/ji are either
ordinary points or accidental singularities of © (z) ; and that the essential
singularities of the group are essential singularities of © (z). Hence we
have the result : —

The series 0 (*) = S (7^ + SO"8"1 H (** \ f *) >

i=o \yiz + f>i/

where the summation extends over the infinite number of members of an assigned
discontinuous group, is a function of z, provided the integer m be > 1 and H(z)
be a rational function of z. The singularities of® are: —

(i), the accidental singularities of H(z) and the points homologous with
them by the substitutions of the group : all these points are acci
dental singularities of @ {z) ;

(ii), the points —&i/yi, which are the points homologous with z = GO by
the substitutions of the group : all these points, if not ordinary
points of © (z), are accidental singularities ; and
(iii), the essential singularities of the group : these lie on the fundamental

circle and they are essential singularities of © (z).

If H (z) had any essential singularity, then that point and all points homo
logous with it by substitutions of the group would be essential singularities
of ©(/). The function ©(2), thus defined, is called* Thetafuchsian by

Poincare.

* Acta Math., t. i, p. 210.

305.] PSEUDO-AUTOMORPHIC PROPERTY 643

If the group belong to the first, the second or the sixth family,
it is known that the circumference of the fundamental circle enters into
the division of the interior of the circle (and also of the space exterior to
the circle) only in so far as it contains the essential singularities of the
group. But if the group belong to any one of the other four families,
then parts of the circumference enter into the division of both spaces.

In the former case, when the group belongs to the set of families,
made up of the first, the second, and the sixth, the circumference of the
fundamental circle is a line over which the series cannot be continued : it
is a natural limit (§81) both for a function existing in the interior of the
circle and for a function existing in the exterior of the circle : but neither
function exists for points on the circumference of the fundamental circle.
The series represents one function within the circle and another function
without the circle.

It has been proved that the area outside the fundamental circle can
be derived from the area inside that circle, by inversion with regard to
its circumference. Hence a function of z, existing only outside the funda

mental circle, can be transformed into a function of , and therefore also

i

of - , existing for points only within the circle. When, therefore, a group
belongs to the first, the second or the sixth family, it is sufficient to consider
only the function defined by the series for points within the fundamental
circle: it will be called the function ®(z).

In the latter case, when the group belongs to the third, the fourth, the
fifth or the seventh families, then parts of the circumference enter into the
division of the plane both without and within the circle. Over these parts
the function can be continued : and then the series represents one (and only
one) function in the two parts of the plane : it will be called the function ® (z).

306. The importance of the function @ (z) lies in its pseudo-automorphic
character for the substitutions of the group, as defined by the property now

to be proved that, if ^ be any one of the substitutions of the group, then

Let

which is, of course, another substitution of the infinite group : then

„ oc* + /3 ,» _7/* + Si'

7V + 8 yz + S '

41—2

644 ZEROS AND SINGULARITIES [306.

(out + 0\ Z (riz + ZiY™ rr (atf*
Hence & — = - H

t=o

thus establishing the pseudo-automorphic character.

This function can evidently be made subsidiary to the construction of
functions, which are automorphic for the group of substitutions, in the same
manner as the o--function in Weierstrass's theory of elliptic functions and
the so-called Theta-functions in the theory of Jacobian and of Abelian
transcendents. But before we consider these automorphic functions, it is
important to consider the zeros and the accidental singularities of a pseudo-
automorphic function such as © (2}.

On the supposition that the function H, which enters as the additive
element into the composition of ©, has only accidental singularities, it has
been proved that all the essential singularities of © lie on the circumference
of the fundamental circle ; and that the accidental singularities of © are,
(i) the points homologous with the accidental singularities of H, and
(ii) the points— Si/7;, which all lie without the circle.

When the function H (2) has one or more accidental singularities within
the fundamental circle, then there is an irreducible point for each of them,
which is an irreducible accidental singularity of © (2-). Hence in the case of
a function which exists only within the circle, the number of irreducible
accidental singularities is the same as the number of (non-homologous} accidental
singularities of H (2) lying within the fundamental circle. If, then, all the
infinities of the additive element H (2) lie without the fundamental circle, and
if the function ©(V) exist only within the circle, then © (2) has no irreducible
accidental singularities : but, in particular cases, it may happen that © (2) is
then evanescent.

When the function H (2} has one or more accidental singularities without
the fundamental circle, then there is an irreducible point for each of them,
this point lying in the fundamental polygon of reference in the space outside
the circle : and this point is an irreducible accidental singularity of © (2),
when © (2) exists both within and without the circle. Further, the point
— S;/7r is an infinity of order 2m : there is a homologous irreducible point
within the polygon of reference without the circle, being, in fact, the
irreducible point which is homologous with 2 = <x> . Hence taking the two
fundamental polygons of reference — one within, for the internal division, and
one without, for the external division, — it follows that in the case of a function,
which exists all over the plane, the number of irreducible accidental singularities

306.] OF A PSETJDO-AUTOMORPHIC FUNCTION 645

is equal to the whole number of accidental singularities of the additive element
H(z), increased by 2m.

307. To obtain the number of irreducible zeros we use the result of
§ 43, Cor. IV., combined with the result just obtained as to the number of
irreducible accidental singularities. A convention, similar to that adopted
in the case of the doubly-periodic functions (§ 115), is now necessary: for if
there be a zero on one side of the fundamental polygon, then the homologous
point on the conjugate side of the polygon is also a zero and of the same
degree : in that case, either we take both points as irreducible zeros and of
half the degree, or we take one of them as the irreducible zero and retain
its proper degree. Similarly, if a corner be a zero, every corner of the cycle
is a zero : so that, if the cycle contain X points and the sum of its angles be

2?r

— , then the corner is common to X/A polygons ; we may regard each of the
A*
corners of the fundamental polygon in that cycle as an irreducible zero, of

degree equal to its proper degree divided by X/i, or we may take only one of
them and count its degree as the proper degree divided by yu, — the just
distribution of zeros common to contiguous polygons being all that is
necessary for the convention — so that the number of zeros to be associated
with the area of each polygon is the same, while no zero is counted in more
than its proper degree. A similar convention applies to the singularities.

With this convention, the excess of the number of irreducible zeros
over the number of irreducible accidental singularities, each in its proper
degree, is the value of

taken positively round the fundamental polygon within the circle when the
function ®(z) exists only within the circle, and round the two fundamental
polygons, within and without the circle respectively, when the function ®(z)
exists over the whole plane.

®'(z}

But should an infinity of ^-~ lie on the curve along which integration

*9(*)

extends, (it will arise through either a zero or a pole of @), then, in order
to avoid the difficulty in the integration and preserve the above convention,
methods must be adopted depending upon the family of the group.

When all the cycles belong to the first sub-category (§ 292), we can
proceed as follows : the general result can be proved to hold in every case.
If an infinity occur on a side, another will occur on the conjugate side, the
two being homologous by a fundamental substitution. A small semicircle is
drawn with the point for centre and lying without the polygon, so that, when
the element of the side is replaced by the semi-circumference, the point
lies within the polygon : the homologous point on the conjugate side is
excluded from the polygon when the element there is replaced by the

646

ZEROS AND SINGULARITIES

[307.

homologous semi-circumference. The subject of integration is then finite
along the modified sides.

A similar process is adopted when a corner is an infinity of . A

small circular arc is drawn so as to have the point included in the polygon
when the arc replaces the elements of the sides at the point : the homologous
circular arcs at all the points in the cycle of the corner will exclude all those
points, also poles, when they replace the elements of the sides at the point.
The subject of integration is then finite everywhere along the modified path
of integration.

First, let the function exist only within the circle. Let AB be any side
of the polygon, A'B' the conjugate side ;
and let

be the corresponding fundamental substi
tution which transforms AB into A'B',
so that £ may be regarded as the variable
along A'B'.

Then we have @ (£) = (<yz + S)2™

Q B

(z),

,
and therefore

dz +

dz.

But as z moves from A to B, % moves from A' to B' (§ 287) : and the latter
is the negative direction of description. Hence, with the given notation, the
sum of the parts of the integral, which arise through the two sides AB
and B'A. is

~ dz, taken along AB ;

z+ -
7

so that, if E denote the required excess, we have

r 7
nm I fi ?

-n III I ' ' -

™, " ~T '

J * + -

7

the new integral being taken along those sides of the polygon which are
transformed into their conjugates by the fundamental substitutions of the
group.

Consider the term which arises through the integration along AB: it is
evidently

307.] OF THETAFUCHSIAN FUNCTIONS 647

d£ 1

JN ow we nave ~ = -, -- srn ,

dz (jz + S)2 '

so that, if M be the magnification in transforming from A to A', and if $a be
the angle through which a small arc is turned, we have at A

- _ . = Me*-.
(yje + 8)-

Evidently <f>a is the excess of the inclination of A'P', that is, of AC' to the
line of real quantities over the inclination of AP, that is, of AC to that line :
and therefore at A

log (7£ + S) = -|logM- !;</>„.
Since the whole integral must prove to be a real quantity, we omit the

97?

parts — — . log M as in the aggregate constituting an evanescent (imaginary)
quantity : hence we have

as the part corresponding to the side AB. In this expression, <j>a is the angle
required to turn AC into a direction parallel to A'C', and <j>b is the angle
required to turn QB, that is, CB into a direction parallel to Q'B', that is,
C'B', both rotations being taken positively. Thus

<f)a = uic\.A'C'-inc\.AC,
<fr, = 27T - incl. EG + incl. B'C' ;
and therefore

<£« - <fr> = - 27r + incl. A'C' - incl. B'C' + incl. EG - incl. AC

= - 2w + c1/ + c1,

where d and c/ are the angles ACS, A C'B' respectively. Hence, if we take
c and c to be the external angles AGE, A C'B' as in the figure, we have

C + d = 2-7T = C' + C/,

and therefore <j)b - (fea — c + c' — ZTT.

The part corresponding to the arc AB in the above integral is therefore

There are no sides of the second kind in the path of integration, because the
function is supposed to exist only within the circle. Therefore the whole
excess is given by

the summation extending over those sides of the polygon, being in number
half of the sides of the first kind, which are transformed into their conjugates
by the fundamental substitutions of the group.

648 EXCESS OF NUMBER OF ZEROS [307.

Draw all the pairs of tangents at the extremities of the bounding arcs
of the fundamental polygon of reference :
then the angles, such as c and c' above,
are internal angles of the rectilinear
polygon formed by the straight lines.
The remaining internal angles of this
new polygon are the angles at which
the arcs cut, which are the angles of
the curvilinear polygon : and therefore
their sum is the sum of the angles in
the cycles, that is, the sum is equal to

o__
where - - is the sum of the angles in Fl&- 125-

one of the cycles. Now let 2n be the number of sides of the first kind in
the curvilinear polygon, so that n is the number of fundamental substitutions
in the group : hence the number of terms in the above summation for E is
n, and therefore

Moreover the rectilinear polygon has 4n sides : and therefore the sum of the

2?r

internal angles is (4w — 2) TT. But this sum is equal to 2 (c + c') + S - — ,

tH

where the first summation extends to the different conjugate pairs and
the second to the different cycles : thus

(471 - 2) 7T = 2 (C + C) +

Therefore E = — mn + m(2n — 1) — m% —

= m ( n — 1 — S — ) ,

where the summation extends over all the different cycles in the fundamental
polygon. Hence for a function, which is constructed from the additive
element H(z) and exists only within the fundamental circle of the group, the
excess of the number of its irreducible zeros over the number of its irreducible
accidental singularities is

m ( n — 1 — 2 — )i

where m is the parametric integer of the function constructed in series. 2n is
the number of sides of the first kind in the fundamental polygon, — is the sum

307.] OVER NUMBER OF SINGULARITIES 64.9

of Hie angles in a cycle of the first kind of corners and the summation extends
to all these cycles.

The number of irreducible accidental singularities has already been
obtained ; it is finite, and thus the number of irreducible zeros is finite.

Secondly, let the function exist all over the plane : then the irreducible
points are (i) points lying within (or on) the boundary of the fundamental
polygon of reference within the fundamental circle and (ii) points lying
within (or on) the boundary of the fundamental polygon of reference without
the fundamental circle, the outer polygon being the inverse of the inner poly
gon with regard to the centre. For such a function the excess of the number
of irreducible zeros over the number of irreducible accidental singularities is
the integral

c~' r\/ \ *j

2-m j 8(jr)

taken positively round the boundaries of both polygons. We shall assume
that there are no zeros and no infinities on the path of integration ; the
result can, however, be shewn to be valid in the contrary case.

For the sides of the internal polygon that are of the first kind the value
of the integral is, as before, equal to

m» — 1 — S —

and for the sides of the external polygon that are of the first kind, the value
is also

( i V l

m (n — 1 — 2)

Let the value of the integral along the sides of the second kind in
the internal polygon be I. Those lines are also sides of the second kind
in the external polygon; but they are described in the sense opposite to
that for the internal polygon, the integral being always taken positively:
hence the value of the integral along the sides of the second kind in the
external polygon is — /.

Hence the excess of the number of irreducible zeros over the number of
irreducible accidental singularities of a function ®(z), which is constructed
from the additive element H(z) and exists all over the plane, is

where the summation extends over all the cycles of the first category of either
(but not both) of the fundamental polygons of reference.

As before, the number of irreducible zeros of such a function is finite,
because the number of irreducible accidental singularities is finite.

650 FUCHSIAN FUNCTIONS [307.

In every case, this excess depends only upon

(i) the parametric integer ra, used in the construction of the series :
(ii) the number of sides, 2/i, of the first kind in the polygon of

reference :
(iii) the sum of the angles in the cycles of the first category.

Ex. Prove that a corner belonging to a cycle of the first category is in general a zero

of order p, such that

p== —m (mod. /z),

where Zir/p. is the sum of the angles in the cycle: and discuss the nature of the corners
which belong to cycles of the remaining categories. (Poincare.)

308. We are now in a position to construct automorphic functions, using
as subsidiary elements the pseudo-automorphic functions which have just
been considered.

For, if we take a couple of these functions, ®x and ®2, associated with a
given infinite group, characterised by the same integer m, and arising through
different additive elements H (z), then we have

+

where — + is any one of the substitutions of the group ; and therefore

z+ 8

that is, the quotient of two such functions is automorphic. Denoting the
quotient by Pn(z)*, we have

.

the automorphic property being possessed for each of the substitutions.

It thus appears that such functions exist: their essential property is
that of being reproduced when the independent variable is subjected to any
of the linear substitutions of the infinite group.

The foregoing is of course the simplest case, adduced at once to indicate
the existence of the functions. The construction can evidently be general
ised : for, if we have any number of functions 01} ..., ®r, ^i, •••> *&s with
characteristic integers m^ ..., mr, n1} ..., ns and all associated with one group

* Poincar6 calls such functions Fuchsian functions : as already indicated (§ 297), I have
preferred to associate the general name automorphic with them. But, because Poincare himself
has constructed one class of such functions by means of series as in the foregoing manner, his
name, if any, should be associated with this class : the symbol Pn (z) is therefore used.

308.] TWO CLASSES OF AUTOMORPHIC FUNCTIONS 651

while constructed from different additive elementary functions H (z), then,
denoting

by Pn (z), we evidently have

22 ni- 2

so that, provided only ^ nq= S mq ,

9=1 J 7=1

the function is automorphic. If we agree to call m, the integer characteristic
of a pseudo-automorphic function, the degree of that function, then the quotient
of two products of pseudo-automorphic functions is automorphic, provided the
products be of the same degree.

There are evidently two classes of automorphic functions : those which
exist all over the plane, and those which exist only within the fundamental
circle. The classes are discriminated according to the composition of the
functions from the subsidiary pseudo-automorphic functions.

When the pseudo-automorphic functions, which enter into the composi
tion of the function, exist all over the plane, then the automorphic function
exists all over the plane. But when the pseudo-automorphic functions, which
enter into the composition of the function, exist only within the fundamental
circle, then the automorphic function exists only within the circle.

309. It is evident that all the essential singularities of an automorphic
function, thus constructed, lie on the fundamental circle. For whether the
pseudo-automorphic functions exist only within that circle or over the whole
plane, all their essential singularities lie on the circumference : so that,
whatever be the constitution of the various subsidiary pseudo-automorphic
functions, all the essential singularities of the automorphic function lie on
the fundamental circle.

Next, the number of irreducible zeros of an automorphic function is equal
to the number of its irreducible accidental singularities. For an irreducible
zero of an automorphic function is either (i) an irreducible zero of a factor
in the numerator or (ii) an irreducible accidental singularity of a factor in
the denominator ; and similarly with the irreducible accidental singularities
of the function. The numerator and the denominator may have common
zeros; this will not affect the result.

First, let the automorphic function exist only within the circle : then
each of its factors exists only within the circle. The space without the circle

652 LEVEL POINTS OF AUTOMORPHIC FUNCTIONS [309.

is not significant for any of the factors of the function, because they do not
there exist. Let elf ..., er, e/, ..., e/ be the excesses of zeros over accidental
singularities for the pseudo-automorphic functions within the fundamental
circle : then

where n and 2 — are the same for all these functions, and

€q' = nq(n — l —

Now the excess of zeros over poles in the denominator becomes, after the
above explanation, an excess of poles over zeros for the automorphic
function : hence, for this automorphic function, the excess of zeros over
accidental singularities is

r s

2^? /
€q~ 2 €q

I 1 V

= I n — L — 2t —
= 0,

r s

by the condition 2 mq = 2 nq. Hence the number of irreducible zeros of

q = \ q = \

the automorphic function is equal to the number of irreducible accidental
singularities.

Secondly, let the automorphic function exist all over the plane; then
all its factors exist all over the plane. For the present purpose, the sole
analytical difference from the preceding case is that each of the quantities e
now has double its former value : and therefore the excess of the number of
zeros over the number of poles is

»f * l\f £ v

2 [n - 1 - 2 — 2 mq- 2 n,

V Pi/ \q = l 9 = 1

which, as before, vanishes. Hence the number of irreducible zeros of the
automorphic function is equal to the number of its irreducible accidental
singularities.

It follows, as an immediate Corollary, that the number of irreducible
points for which an automorphic function assumes a given value is equal to
the number of its irreducible accidental singularities. For

Pn(z)-A,

where A is a constant, is an automorphic function : the number of its
irreducible accidental singularities is equal to the number of its irreducible
zeros, that is, it is equal to the number of irreducible points for which
Pn(z) assumes an assigned value.

309. J DIFFERENT FUNCTIONS FOR ONE GROUP 653

Moreover, each of these numbers is finite : for the number of irreducible
zeros and the number of irreducible accidental singularities of each of the
component pseudo-automorphic factors is finite, and there is only a finite
number of these factors in the automorphic function. The integer, which
represents each number, will evidently be as characteristic of these functions
as the corresponding integer was of functions with linear additive periodicity.

Note. The preceding method, due to Poincare', of expressing the pseudo-
automorphic functions as converging infinite series of functions of the
variable, is not the only method of obtaining such functions. It was
shewn that uniform analytical functions can be represented either as
converging series of powers or as converging series of functions or as
converging products of primary factors, not to mention the (less useful)
forms intermediate between series and products. The representation of
automorphic functions as infinite products of primary factors is considered
in the memoirs of Von Mangoldt and Stahl, already referred to in § 297.

310. Let P%(Y), Pnz(z}, say Pl and P2, be two automorphic functions
with the same group, constructed with the most general additive elements :
and let the number of irreducible zeros of the former be KI} and of the
latter be K2.

Then for an assigned value of P^ there are ^ irreducible points : P2 has a
single value for each of these points, and therefore it has ^ values altogether
for all the points, that is, it has /^ values for each value of Px. Similarly, Pj
has K2 values for each value of P2. Hence there is an algebraical relation
between P! and P2 of degree K2 in Px and of degree ^ in P2, which may be
expressed in the form

F12(Pl,P2) = 0.

Let Pn (z), say P, be any other uniform automorphic function, having the
same group as P1 and P2 : and let K be the number of its irreducible zeros.
Then we have an algebraical equation

Fl(P,Pl) = 0,
which is of degree «„ in P and of degree K in P1 ; and another equation

P,(P,P2) = 0,

which is of degree Ki in P and of degree K in P2. The last two equations
coexist, in virtue of the relation

F12(P1}PZ) = 0

satisfied by P1 and P2. Since Fl=0 = F2 coexist, the ordinary theory of
elimination leads to the result that the uniform function P can be expressed
rationally in terms of Pj and P2, so that we have the theorem that every
automorphic function associated with a given group can be expressed rationally
in terms of two general automorphic functions associated with that group : and
between these two functions there exists an irreducible algebraical relation.

654 ALGEBRAICAL RELATIONS [310.

The class (§ 178) of this algebraical relation can be obtained as follows.
Let N denote the class of the group, determined as in § 293 : then the funda
mental polygon of reference, if functions exist only within the circle, or the
two fundamental polygons of reference, if functions exist over the whole
plane, can be transformed into a surface of multiple connectivity 2N+ 1. The
automorphic functions are functions of uniform position on this surface ; and
hence, as in Riemann's theory of functions, the algebraical relation between
two general uniform functions of position, that is, between two general auto
morphic functions is of class N, where N is the class of the group *.

It is now evident that the existence-theorem and the whole of Riemann's
theory of functions can be applied to the present class of functions, whether
actually automorphic or only pseudo-automorphic. There will be functions
of the same kinds as on a Riemann's surface : the periods will be linear
numerical multiples of constant quantities acquired by a function when its
argument moves from any position to a homologous position or returns to its
initial position. There will be functions everywhere finite on the surface,
that is, finite for all values of the variable z except those which coincide with
the essential singularities of the group. The number of such functions,
linearly independent of one another, is N ; and every such function, finite for
all values of z except at the essential singularities, can be expressed as a
linear function of these N functions with constant coefficients and (possibly)
an additive constant. And so on, for other classes of functions^.

311. Because Pn (z) is an automorphic function, we have

D
P

and therefore, as aS — /3y = 1,

Hence, if © (z) be a pseudo-automorphic function with m for its character
istic integer, so that

,
we have

* It may happen that, just as in the general theory of algebraical functions, the class of the
equation between two particular automorphic functions may be less than N : thus one might
be expressed rationally in terms of the other. The theorems are true for functions constructed
in the most general manner possible.

t The memoirs by Burnside, quoted in § 297, develop this theory in full detail for the group
which has its (combined) polygons of reference bounded by 2n circles with their centres on the
axis of real quantities, the group being such that the pseudo-automorphic functions exist over the
whole plane.

311.] DERIVATIVES OF AUTOMORPHIC FUNCTIONS 655

that is, ® (z} \Pn' (z)}~m is an automorphic function. Such a function can be
expressed rationally in terms of Pn (z) and some other function, say of P and
Q : hence the general type of a pseudo-automorphic function with a charac
teristic integer m is

where f is a rational function.

COROLLARY. Two automorphic functions P and Q, belonging to the same
group, are connected by the equation

dz d

For evidently unity is the characteristic integer of the first derivative of an
automorphic function.

This equation can be changed to

where f is a rational function : moreover P and Q are connected by an
equation

which is an algebraical rational equation, and can evidently be regarded as
an integral of the above differential equation of the first order, all trace of
the variable z having disappeared. Evidently the form of f is given by

A • A \- az + @, , D (ajs+\, TT/^S

Again, denoting ----- *7 by f, and Pn ( --- ~ J by II (£), we have

say

so that

IT" en")2 r P"

I ~ if

IT" CTT")2 VP"' fP"12l

and therefore ^ - 1 J^J . (7^ + S)< ^ - f j^j J

whence

where {P, z] is the Schwarzian derivative. It thus appears that, if P be an

(>5() DIFFERENTIAL EQUATIONS [311.

autoniorphic function, then {P, z\ P'~J is a function automorphic for the same
group.

But between two automorphic functions of the same group, there subsists
an algebraical equation : hence there is an algebraical equation between P
and {P, z\ P'~2, that is, P (z), an automorphic function of z, satisfies a
differential equation of the third order, the degree of which is the integer
representing the number of irreducible zeros of P and the coefficients of which,
where they are not derivatives of P, are functions of P only and not of the
indepen den t variable.

This equation can be differently regarded. Take

yi = . yt =
then it is easy to prove that

1 ^ _ 1 d*y* - 1

a * r

The last fraction has just been proved to be an automorphic function of z\
and therefore it is rationally expressible in terms of P and any other general
function, say Q, automorphic for the group. Then yl and y2 are independent
integrals of the equation

where Q and P are connected by the algebraical equation

F(P,Q) = 0.
Conversely, the quotient of two independent integrals of the equation

where Q and P are connected by the algebraical equation

can be taken as an argument of which P and Q are automorphic functions :
the class of the equation F=0 is the class of the infinite group of substitutions
for which P and Q are automorphic*.

Ex. One of the simplest set of examples of automorphic functions is furnished by
the class of homoperiodic functions (§ 116). Another set of such examples arises in the
triangular functions, discussed in § 275 ; they are automorphic for an infinite group, and
the triangles have a circle for their natural limit. A third set is furnished by the polyhedral
functions (§§ 276—279).

As a last set of examples, we may consider the modular-functions which were
obtained by a special method in § 303.

* Klein remarks (Math. Ann., t. xix, p. 143, note 4) that the idea of uniform automorphic
functions occurs in a posthumous fragment by lliouiaun (G^x. HYrAv, number xxv, pp. 413 — 410).
ll may also be pointed out that the association of such functions with the linear differential
equation of the second order is indicated by Kiemann.

311.] MODULAR-FUNCTIONS 657

First, we consider them in illustration of the algebraical relations between functions
automorphic for the same group. It follows, from the construction of the group and the
relation of c to w, that, in the division of the plane by the group with Uw and Vw for its
fundamental substitutions, where

W

Uw=w + 2, Vw = - — — - ,
1 - Zw '

there is only a single point in each of the regions for which c has an assigned value ; hence,
regarding c as an automorphic function of w, the number K (§ 310) is unity. If there bo
any other function C of w, automorphic for this group, then between C and c there is an
algebraical relation of degree in C equal to the number K for c, that is, of the first degree
in C. Hence every function automorphic for the group, whose fundamental substitutions
are U and V, where

=-—,
1 - "2 w
is a rational algebraical function of c.

In the same way, it can be inferred that every function automorphic for the group,

whose fundamental substitutions are

is a rational, algebraical, function ofcc'; and that every function automorphic for the group,
whose fundamental substations are

that is, automorphic for all substitutions of the form ® ~j , where a, b, c, d are real

CIO ~\~ ('

integers, such that ad-bc = l, is a rational algebraical function o/ ./= ' C . ~ c '" <! .

c (c — 1)

Secondly, in illustration of the general theorem relating to the differential equation
of the third order which is characteristic of an automorphic function, we consider the

quantity c as a function of the quotient of the quarter-periods. Let z denote *->- : then

lxvau.se every function automorphic for the same group of substitutions as c is a rational
function of c, we have

{c z}

^72~-= rational function of c ;

C

and therefore, by a property of the Schwarzian derivative,

{z, c} = - same rational function of c.

By known formula) of elliptic functions, it is easy to shew that

1-c + c2

thus verifying the general result.

Similarly, it follows that |-y , 01, where 0 = ce', is a rational function ofcc', the actual

value being given by

(iK' } _ 1-50 + 1602
\K ' ) ~ 20* (1-40)3 !

f ' It*"1 ~\

and that \ -^ ,J\ is a rational function of «/, the actual value being given by

(iK' }_ 16J* -123 J - :v,\( )
1 K '* J 27*^(47- 27)* '
In this connection a memoir by Hurwitz* may bo consulted.

* Math. Ann., t. xxxiii, (1889), pp. 345—352.

v- 42

658 CONCLUSION [311.

The preceding application to differential equations is only one instance
in the general theory which connects automorphic functions with linear
differential equations having algebraical coefficients. This development
belongs to the theory of differential equations rather than to the general
theory of functions: its exposition must be reserved for another place.

Here my present task comes to an end. The range of the theory of
functions is vast, its ramifications are many, its development seems illimit
able : an idea of its freshness and its magnitude can be acquired by noting
the results, and appreciating the suggestions, contained in the memoirs of
the mathematicians who are quoted in the preceding pages.

GLOSSAEY

OF TECHNICAL TERMS USED IN THE THEORY OF FUNCTIONS.

(The numbers refer to the pages, where the term occurs for the
first time in the book or is defined.)

Abbildung, conforme, 11.

Absoluter Betrag, 3.

Accidental singularity, 16, 53.

Addition- theorem, algebraical, 297.

Adelphic order, 317.

Algebraical addition-theorem, 297.

Algebraical function, rational, 70.

Algebraical function determined by an equation,

161.

Amplitude, 3.

Analytical curve, 409, 423, 530.
Analytic function, monogenic, 56.
Argument, 3.

Argument and parameter, interchange of, 451.
Arithmetic mean, method of the, 408.
Ausscrwesentliche singuldre Stelle, 53.
Automorphic functions, 582, 619.

Betrag, absoluter, 3.
Bien defini, 161.
Bifacial surface, 325.
Boundary, 322.
Branch, 15.
Branch-line, 339.
Branch-point, 15, 154.
Branch-section, 339.

Canonical resolution of surface, 355.

Categories of corners, cycles, 592, 596.

Circle, discriminating, 111.

Circuit, 327.

Class (of connected surface), 324.

Class of doubly-periodic function of second

order, 223.

Class of equation, 349.
Class of group, 608.
Class of singularity, 147.

Class of tertiary-periodic function, 288.

Class of transcendental integral functions, 89.

Combination of areas, 425.

Compound circuit, 327.

Conformal representation, 11.

Conforme Abbildung, 11.

Congruent figures, 517, 591.

Conjugate edges, 592.

Connected surface, 312.

Connection, order of, 317.

Connectivity, 317.

Constant modulus for cross-cut, 377.

Contiguous regions, 591.

Continuation, 55.

Continuity, region of, 55.

Continuous substitution, 584.

Convergence, uniform unconditional, 127.

Convexity of normal polygon, 594.

Corner of region, 591.

Coupure, 140, 186.

Critical point, 15.

Cross-cut, 314.

Cross-line, 339.

Cycles of corners, 593.

Deficiency, 356.

Deformation of loop, 357.

Deformation of surface, 333.

Degree of pseudo-automorphic function, 651.

Derivative, Schwarzian, 529.

Dihedral group, 623.

Diramazione, punto di, 15.

Dirichlet's principle, 408.

Discontinuity, polar, 16.

Discontinuous groups, 584.

Discontinuous substitution, 584.

Discrete substitution, 584.

42—2

660

GLOSSARY OF TECHNICAL TERMS

Discriminating circle, 111.

Domain, 52.

Double (or fixed) circle of elliptic substitu
tion, 613.

Doubly-periodic function of first, second, third,
kind, 273, 274.

Edge of region, 591.

Edges of cross-cut, positive and negative, 375.

Eindndrig, 15.

Eindeutig, 15.

Einfach zusammenhangend, 313.

Element, 56.

Element of doubly-periodic function of third

kind, 291, 293.
Elementary integral of the second kind, third

kind, 396, 402.
Elliptic substitution, 517.
Equivalent homoperiodic functions, 220.
Essential singularity, 17, 53.
Existence-theorem, 369, 405.

Factor, primary, 82.

Factorial functions, 464.

Families of groups, 606.

Finite groups, 586.

First kind, doubly-periodic function of the,

273.

First kind of Abelian integrals, 394.
Fixed (or double) points of substitution, 514.
Fortsetzung , 55.

Fractional factor for potential function, 422.
Fractional part of doubly-periodic function,

220.

Fuchsian functions, 619.
Fuchsian groups, 606.
Fundamental circle for group, 603.
Fundamental loops, 360.
Fundamental parallelogram, 200.
Fundamental polyhedron (of reference for

space), 615.
Fundamental region (of reference for plane),

591.
Fundamental substitutions, 583.

Gattung (kind of integral), 394.

Genere, 89.

Genere (class of connected surface), 324.

Geschlecht, 324.

Genre (applied to singularity), 148.

Genre (applied to transcendental integral

functions), 89.

Genre (class of connected surface), 324.
Giramento, punto di, 15.
Gleichmassig unbedingt convergent, 127.

Gleichverzweigt, 371.
Grenze, natiirliche, 129.
Grenzkreis, 111.
Group of substitutions, 582.
Grundzahl, 317.

Hauptkreis, 603.
Holomorphic, 15.
Homogeneous substitutions, 622.
Homographic transformation, or substitution,

512.

Homologous (points), 200.
Homoperiodic, 224.
Hyperbolic substitution, 517.

Improperly discontinuous groups, 585.

Infinitesimal substitution, 522.

Infinity, 16.

Integrals of the first kind, second kind, third

kind, Abelian, 394, 396, 400.
Interchange of argument and parameter, 451.
Invariants of elliptic functions, 250.
Inversion-problem, 455.
Irreducible circuit, 327.
Irreducible (point), 199, 200.
Isothermal, 576.

Kleinian functions, 619.
Kleinian groups, 610.

Lacet, 153.

Lacunary functions, 141.

Ligne de passage, 339.

Limit, natural, 129.

Limitrophe, 591.

Linear substitution, 512.

Loop, 153.

Loop-cut, 315.

Loxodromic substitution, 517.

Mehrdeutig, 15.

Mehrfach zusammenhangend, 314.

Meromorphic, 16.

Modular-function, 633.

Modular group, 587.

Modulus, 3.

Modulus for cross-cut, constant, 377.

Modulus of periodicity (cross-cut), 377.

Monadelphic, 313.

Monodromic, 15.

Monogenic, 14.

Monogenic analytic function, 56.

Monotropic, 15.

Multiform, 15.

Multiple circuit, 327.

GLOSSARY OF TECHNICAL TERMS

661

Multiple connection, 314.
Multiplicateurs, fonctions a, 464.
Multiplier of substitution, 515.

Natural limit, 129.

Natilrliche Grenze, 129.

Negative edge of cross-cut, 375.

Niveaupunkte (points where a function acquires

any, the same, value, 227).
Non-essential singularity, 53.
Normal (connected) surface, 334.
Normal form of linear substitution, 582.
Normal function of first kind, second kind,

third kind, 446, 448, 450.
Normal polygon for substitutions, 594.

Order of a doubly-periodic function, 220.
Order, of connection, adelphic, 317.
Ordinary point, 52.
Orthomorphosis, 11.

Parabolic substitution, 517.

Parallelogram, fundamental or primitive, 200,

206.

Path of integration, 18.
Period, 198.

Periodicity for cross-cut, modulus of, 377.
Polar discontinuity, 16.
Pole, 16, 53.
Polyadelphic, 314.
Polyhedral functions, 575.
Poly tropic, 15.

Positive edge of cross-cut, 375.
Potential function, 407.
Primary factor, 82.
Prim/unction, 82.
Primitive parallelogram, 206.
Properly discontinuous groups, 585.
Pseudo-periodicity, 256, 259, 273, 274.

Querschnitt, 314.

Ramification (of Riemann's surface), 349.

Ramification, point de, 15.

Rational algebraical function, 70.

Rational points, 141.

Rational transcendental function, 70.

Real substitutions, 517.

Reconcileable circuits, 327.

Reducible circuit, 327.

Reducible (point), 199, 200.

Region of continuity, 55.

Regular, 16, 52.

Regular singularities, 163.

Representation conforme, 11.

Residue, 42.

Resolution of surface, canonical, 355.

Retrosection, 315.

Riemann's surface, 336.

Root, 16.

Riickkehrschnitt, 315.

Schleife, 153.

Schwarzian derivative, 529.

Second kind, doubly-periodic function of the,
274.

Second kind of Abelian integrals, 396.

Secondary-periodic functions, 275.

Section, 140, 186.

Section (cross-cut), 314.

Sheet, 336.

Simple branch-points, 174.

Simple circuit, 327.

Simple connection, 313.

Simple curve, 21.

Simple cycle of loops, 360.

Simple element of positive class for tertiary-
periodic function, 291.

Singular point, 16.

Singularity, accidental, 16, 53.

Singularity, essential, 17, 53.

Species of singularity, 148.

Sub-categories of cycles, 607.

Substitution, homogeneous, 622.

Substitution, linear or homographic, 512.

Synectic, 15.

Taglio trasversale, 314.
Tertiary-periodic functions, 275.
Tetrahedral group, 625.
Thetafuchsian function, 642.
Third kind, doubly-periodic function of the, 274.
Third kind of Abelian integral, 400.
Transcendental function, rational, 70.
Trasversale, 314.

Umgebung, 52.
Unifacial surface, 325.
Uniform, 15.

Verzweigungschnitt, 339.
Verzweigungspunkt, 15.

Wesentliche singuldre Stelle, 53.
Winding-point, 346.
Winding-surface, 346.
Windungspunkt, 15.

Zero, 16.

Zusammcnhdngend, einfach, mehrfach, 313, 314.

INDEX OF AUTHORS QUOTED.

(The numbers refer to the pages.)

Abel, 456.

Anissimoff, 111.

Appell, 145, 295, 296, 464 et seq.

Argand, 2.

Ascoli, 409.

Beltrami, 530, 532, 533.

Bianchi, 617, 618.

Biehler, 295.

Biermann, 55, 297.

Bolza, 583.

Bonnet, 500.

Borchardt, 218.

Brill, 356, 367, 464.

Brioschi, 275, 281.

Briot, 456, 464.

Briot and Bouquet, vii, 23, 39, 41, 168, 173,

208, 210, 218, 230, 456, 482, 489, 490.
Burnside (W.), 117, 247, 355, 406, 524, 536, 558,

620, 640, 654.
Burnside (W. S.) and Panton, 390.

Cantor, 147.

Casorati, 2, 23, 359.

Cathcart, 6.

Cauchy, v, vii, 23, 27, 43, 51, 61, 180, 312.

Cayley, 2, 11, 74, 356, 476, 482, 509, 510, 515,

529, 530, 533, 537, 549, 575, 576, 579, 620,

622.

Cesaro, 92.

Christoffel, 538, 541, 549.
Chrystal, vi, 2, 6, 48, 170, 184.
Clebsch, 174, 209, 356, 359, 361, 363, 364, 367,

403, 455, 456.
Clifford, 333, 361.

Darboux, 20, 46, 57, 69, 538, 549.
Dedekind, 633, 637.
De Sparre, 92.

Dingeldey, 335.
Dini, vi.

Du Bois-Eeymond, 133.
Durege, 54, 316, 335.
Dyck, 335, 583, 585.

Eisenstein, 85, 87.
Enneper, 637.

Falk, 201.

Floquet, 282.

Fredholm, 46.

Fricke, vii, 129, 403, 460, 462, 464, 511, 512, 573.

Frobenius, 266, 275, 281.

Frost, 333.

Fuchs, 111, 482, 637.

Galois, 582.

Gauss, 2, 11, 84, 408, 491, 496, 500.

Gordan, 174, 209, 359, 361, 367, 403, 455, 456,

586.

Goursat, 144, 188, 189, 295, 546, 549, 618.
Green, 408.
Greenhill, 193.
Giinther, 464.

Guichard, 104, 147, 217, 218.
Gutzmer, 46.
Gylddn, 126.

Halphen, 85, 263, 266, 275, 185, 295, 296.

Hankel, 129, 189.

Harnack, 6, 10, 409.

Heine, 189.

Henrici, 409.

Hermite, vii, 20, 77, 84, 92, 112, 140, 185,

186, 188, 189, 257, 275, 277, 279, 286, 295,

456, 464, 476, 633, 637.
Herz, 500.
Hobson, 6, 83.

INDEX OF AUTHORS QUOTED

663

Holder, 54.

Hofmann, 366.

Holzmiiller, 2, 345, 512.

Homen, 144.

HoUel, 2.

Humbert, 456, 464.

Hurwitz, 406, 585, 588, 637, 657.

Jacobi, 88, 189, 194, 200etseq., 238, 456, 500, 501.
Jordan, 35, 129, 188, 193, 195, 583.

Kirchhoff, 514.

Klein, vii, viii, 129, 334, 335, 369, 403, 408,

456, 460, 462, 464, 501, 511, 512, 517, 549,

573, 583 et seq., 619 et seq.
Korkine, 497, 500.
Konigsberger, 230, 456.
Kopcke, 136.
Krause, 295.
Krazer, 456.

Lachlan, 557, 562.

Lagrange, 497, 499, 500.

Laguerre, 89, 91, 92.

Lam6, 281, 576.

Laurent, 43, 47, 49, 50, 51, 213.

Legendre, 194.

Lerch, 136.

Lhuilier, 325.

Lindemann, 356, 464.

Liouville, 161, 210, 218, 230.

Lippich, 316, 335.

Love, 543.

Liiroth, 359, 361.

Mathews, 617.

Maxwell, 408.

Meyer, 576.

Michell, 541.

Mittag-Leffler, vii, 57, 112 et seq., 147, 275,

278, 279.
Mobius, 325, 512, 573.

Nekrassoff, 111.

Netto, 583.

Neumann, vii, 5, 6, 37, 153, 161, 316 et seq.,

347, 354, 408, 409, 456, 464, 468.
NSther, 356, 464.

Painleve, 140.

Phragm^n, 297, 407, 538.

Picard, 54, 141, 282, 296, 464, 617.

Pincherle, 146.

Pochhammer, 188.

Poincare, viii, 92, 141, 144, 295, 297, 512,

518—520, 523, 583 et seq., 619 et seq.
Poisson, 408.
Pringsheim, 137, 201.
Prym, 353, 354, 369, 409, 456, 464.
Puiseux, 168.

Baffy, 482.

Eausenberger, 295, 586.

Eiemann, v, vii, 8, 10, 14, 21; 133, 180, 186,

312 et seq., 336 et seq., 368 et seq., 408, 409,

447, 456, 459, 464, 500, 526, 656.
Eiemann, J. , 409.
Bitter, 620.
Boch, 459, 464.
Bouch6, 46.

Salmon, 356, 367.

Schlafli, 538.

Schlesinger, 620.

Schlomilch, 2.

Schonflies, 618.

Schottky, 525, 619.

Schroder, 137.

Schwarz, vii, 13, 57, 129, 136, 297, 308, 405

et seq., 490, 505, 506, 526 et seq.
Seidel, 137.
Serret, 583.
Siebeck, 579.
Stahl, 619, 653.
Stickelberger, 266, 456.
Stieltjes, 144.
Stokes, 408.
Stolz, vi.

Tannery, vi, 137.

Teixeira, 145.

ThomaB, 463.

Thomson (Lord Kelvin), 408.

Todhunter, 20.

Vivanti, 92.

Von Mangoldt, 619, 653.

Von der Miihll, 500, 576.

Weber, 189, 511, 619, 633, 637.

Weierstrass, v, vii, 14, 44, 53, 54, 55, 57, 74

et seq., 97 et seq., 112 et seq., 238, 254, 297,

311, 455, 456.
Weyr, 84.
Wiener, 136.
Williamson, 20, 40.
Witting, 93.

GENERAL INDEX.

(The numbers refer to the pages.)

Abelian transcendental functions, arising by
inversion of functions of the first kind on
a Eiemann's surface, 455 ;

Weierstrass's form of, 456.
Accidental singularities, 16, 53, 64 ;

must be possessed by uniform function,

64;

form of function in vicinity of, 64 ;
are isolated points, 65 ;
number of, in an area, 67, 68, 72 ;
if at infinity and there be no other
singularity, the function is algebraical
polynomial, 69;

if there be a finite number of, and no
essential singularity, the uniform
function is rational, algebraical and
meromorphic, 71.

Addition-theorem, for uneven doubly-periodic
function of second order and second class,
247;

for Weierstrass's ^-function, 262 ;
partial form of, for the a-function and

the f-function, 261 ;
definition of algebraical, 297 ;
algebraical, is possessed by algebraical
functions, 297;

by simply-periodic functions, 298 ;
by doubly-periodic functions, 299 ;
function which possesses an algebraical, is
either (i) algebraical, 300 ;
or (ii) simply-periodic, 303, 305 ;
or (iii) doubly-periodic, 307 ;
satisfies a differential equation be
tween itself and its first derivative,
308;

condition that algebraical equation be
tween three variables should express,
310;

form of, when function is uniform, 311 ;
reference to binomial differential equa
tions, 490.

Algebraic equation between three variables
should express an addition-theorem, condi
tion that, 310 ;

Algebraic equation, defining algebraic multi
form functions, 161 (see algebraic function) ;
class of, 349 ;
for any uniform function of position on

a Eiemann's surface, 371.
Algebraic function, rational integral, 70.
Algebraic (multiform) functions defined by
algebraical equation, 161 ;
branch-points of, 162 ;
infinities of, are singularities of the
coefficients, 163 ;

graphical method for determination

of order of, 164 ;
branch-points of, 168 ;
cyclical arrangements of branches round

a branch-point, 171 ;
when all the branch-points are simple,

174;
in connection with Eiemann's surface,

338.

Algebraic function on a Eiemann's surface,
integrals of, 387 ;

integrals of, everywhere finite, 388 ;

number of, in a special case, 388 ;
when all branch-points are simple, three

kinds of integrals of, 389 ;
infinities of integrals of, 390, 393 ;
branch-points of integrals of, 393.
Algebraic functions on a Eiemann's surface,
constructed from normal elementary func
tions of second kind, 457 ;

smallest number of arbitrary infinities
to render this construction possible,
457;

Eiemann-Eoch's theorem on, 459 ;
smallest number of infinities of, which,
except at them, is everywhere uniform
and continuous, 460 ;

GENERAL INDEX

665

which arise as first derivatives of func
tions of first kind, 460 ;
are infinite only at branch-points,

460;
number of infinities of, and zeros

of, 461 ;

most general form of, 461 ;
determined by finite zeros, 462 ;
determine a fundamental equation for a

given Eiemann's surface, 462 ;
relations between zeros and infinities of,

468.
Algebraic isothermal curves, families of, 576

et seq. (see isothermal curves).
Algebraic relation between functions automor-
phic for the same infinite group, 653 ;

class of, in general, 654.
Analytic function, monogenic, 56.
Analytical curve, 409, 423, 530 ;

represented on a circle, 423;
area bounded by, represented on a half-
plane, 530 ;

consecutive curve can be chosen at

will, 531.

Analytical test of a branch-point, 157.
Anchor-ring conformally represented on plane,

501.

Anharmonic group of linear substitutions, 620.
Anharmonic function, automorphic for the an-

harmonic group, 620.
Appell's factorial functions, 464 (see factorial

functions).

Area, simply connected, can be represented
conformally upon a circle with unique cor
respondence of points, by Eiemann's theorem,
526;

form of function for representation on a

plane, 528, 540 ;
form of function for representation on a

circle, 529 ;
bounded by analytical curve represented

on half -plane, 530 ;

bounded by cardioid on half-plane, 536 ;
of convex rectilinear polygon, 537 et

seq. (see rectilinear polygon) ;
bounded by circular arcs, 549 et seq. (see

curvilinear polygon).

Areas, combination of, in proof of existence-
theorem, 425.

Argument (or amplitude) of the variable, 3.
Argument of function possessing an addition-
theorem, forms of, for a value of the function,
300 et seq.

Argument and parameter of normal elementary
function of third kind, 453.

Automorphic function, 619 ; <i"" £ 2—

constructed for infinite group in pseudo-
automorphic form, 638 et seq. (see
thetafuchsian functions) ;

expressed as quotient of two theta
fuchsian functions, 651 ;

its essential singularities, 651 ;

number of irreducible zeros of, is the
same as the number of irreducible
accidental singularities, 651 ;

different, for same group are connected
by algebraical equation, 653 ;

class of this algebraical equation in
general, 654;

connection between, and general linear
differential equations of second order,
656;

modular-functions as examples of, 657.

Barriers, impassable, in connected surface, 313 ;
can be used to classify connected sur
faces, 314 ;

changed into a cut, 314.
Bifacial Surfaces, 325, 333.
Binomial differential equations of first order
when the integral is uniform, with the
various classes of integrals, 482 et seq.
Boundary of region of continuity of a function
is composed of the singularities of the
function, 57.
Boundary, defined, 322 ;

assigned to every connected surface, 314,

322, 329 ;

edges acquired by cross-cut and loop-
cut, 315;
of simply connected surface is a single

line, 323 ;
effect of cross-cut on, 323 ;

and of loop-cut on, 324.
Boundary conditions for potential function,

410 (see potential function).
Boundary values of potential function for a
circle, 414 ;

may have limited number of finite dis
continuities, 417 ;
include all the maxima and the minima

of a potential function, 422.
Boundaries of connected surface, relation be
tween number of, and connectivity, 324.
Branch-lines, are mode of junction of the sheets
of Eiemann's surfaces, 339 ;
properties of, 340 et seq. ;
free ends of, are branch-points, 340 ;
sequence along, how affected by branch
points, 341 ;

666

GENERAL INDEX

system of, for a surface, 341 ;

special form of, for two-sheeted surface,

344;

when all branch-points are simple,

356;
number of, when branch-points are

simple, 364.
Branches of a function, denned, 15 ;

affected by branch-points, 151 et seq.;

obtained by continuation, 151 ;

are uniform in continuous regions where

branch-points do not occur, 155 ;
which are affected by a branch-point,

can be arranged in cycles, 156 ;
restored after number of descriptions of

circuit round branch-point, 157 ;
analytical expression of, in vicinity of

branch- point, 158;
number of, considered, 159 ;
of an algebraic function, 161 (see alge
braic function) ;
a function which has a limited number

of, is a root of an algebraic equation,

175.
Branch-points, denned, 15, 154 ;

integral of a function round any curve

containing all the, 37 ;
effect of, on branches, 149, 151, et seq.;
analytical test of, 157 ;
expression of branches of a function in

vicinity of, 158 ;

of algebraic functions, 162, 168 ;
simple, 174, 355;
number of simple, 175 ;
are free ends of branch -lines, 340 ;
effect of, on sequence of interchange

along branch-lines, 341 ;
joined by branch-lines when simple, 344;
deformation of circuit on Eiemann's

surface over, is impossible, 350 ;
circuits round two, are irreducible, 350 ;
number of, when simple, 356 ;
in connection with loops, 357 (see

loops) ;
canonical arrangement of, when simple,

364.

Canonical form of complete system of simple
loops, 361 ;

Eiemann's surface, 365 ;

resolved, 366.

Canonical resolution of Eiemann's surface, 355.
Cardioid, area bounded by, represented on strip
of plane, 535 ;

on a circle, 536.

Categories of corners, 592 (see corners).
Cauchy's theorem on the integration of a holo-
morphic function round a simple curve, 23 ;
and of a meromorphic function, 27 ;
on the expansion of a function in the

vicinity of an ordinary point, 43.
Circle, areas of curves represented on area of :
exterior of ellipse, 501 ;
interior of ellipse, 504 ;
interior of rectangle, 502, 544 ;
interior of square, 503, 545 ;
exterior of square, 545 ;
exterior of parabola, 505 ;
interior of parabola, 506 ;
half-plane, 506 ;
interior of semicircle, 507 ;
infinitely long strip of plane, 508 ;
any circle, by properly chosen linear

substitution, 514 ;
any simply connected area, by Eiemann's

theorem, 526 ;
interior of cardioid, 536 ;
interior of regular polygon, 548 (Ex.).
Circuits, round branch-point, effect of, on
branch of a function, 153, 155 ;

restore initial branch after number of

descriptions, 157 ;
on connected surface, 327 ;
reducible, irreducible, simple, multiple,

compound, reconcileable, 327 ;
represented algebraically, 328 ;
complete system of, contains unique

number of circuits, 328 ;
drawn on a simply connected surface are

reducible, 329 ;
number in complete system for multiply

connected surface, 330 ;
cannot be deformed over a branch-point

on a Eiemann's surface, 350.
Circular functions obtained, by integrating
algebraical functions, 191 ;

on a Eiemann's surface, 380.
Class of, algebraic equation associated with a
Eiemann's surface, 349 ;

between automorphic functions, 654 ;
connected surface, 324 ;
essential singularity, 147 ;
Fuchsian group, 608 ;
Laguerre's criterion of, 91 ;
Eiemann's surface, 349 ;
simple function of given class, 91 ;
tertiary-periodic function, positive, 288 ;

negative, 291 ;

transcendental integral function, as de
nned by its zeros, 89.

GENERAL INDEX

667

Classes of doubly-periodic functions of the

second order are two, 223.
Closed cycles of corners in normal polygon for

division of plane, 596 (see corners).
Combination of areas, in determination of

potential function, 425.
Complex variable defined, 1 ;

represented on a plane, 2 ;

and on Neumann's sphere, 4.
Compound circuits, 327.

Conditions that one complex variable be a func
tion of another, 7.

Conformal representation of planes, established
by functional relation between variables, 11;
magnification in, 11 ;
used in Schwarz's proof of existence-
theorem, 423 ;

most generalform of relation that secures,
is relation between complex variables,
496;

examples of, 501 et seq.

Conformal representation of surfaces is secured
by relation between complex variables in the
most general manner, 492 ;

obtained by making one a plane, 495 ;
of surfaces of revolution on plane, 496 ;
of sphere on plane, 497 ;

Mercator's and stereographic projec
tion, 498 ;

of oblate spheroid, 500 ;
of ellipsoid, 500 ;
of anchor-ring, 501 ;
Biemann's general theorem on, 526 ;
form of function for, on a plane, 528 ;

on a circle, 529.
Congruent regions by linear substitutions, 517,

591.

Conjugate edges of a region, 592 (see edges).
Connected surface, supposed to have a boundary,
314, 322, 329 ;

to be bifacial, 325 ;

divided into polygons, Lhuilier's theorem

on, 325 ;
geometrical and physical deformation of,

333;

can be deformed into any other connected
surface of the same connectivity having
the same number of boundaries, if both
be bifacial, 334 ;
Klein's normal form of, 334.
Connection of surfaces, defined, 312 ;
simple, 313 ;

definition of, 315 ;
multiple, 314 ;

definition of, 315 ;

affected by cross-cuts, 319 ;
by loop-cuts, 320 ;
and by slit, 321.

Connectivity, of surface defined, 317 ;
affected by cross-cuts, 319 ;
by loop-cuts, 320 ;
by slit, 321 ;

of spherical surface with holes, 321 ;
in relation to irreducible circuits, 330 ;
of a Eiemann's surface, with one boun
dary, 347 ;

with several boundaries, 350.
Constant, uniform function is, everywhere if

constant along a line or over an area, 59.
Constant difference of integral, at opposite
edges of cross-cut, 375 ;

how related for cross-cuts that meet, 376 ;
for canonical cross-cuts, 377 (see

moduli of periodicity).
Contiguous regions, 591.

Continuation, of function by successive domains,
54;

Schwarz's symmetric, 57 ;

of function with essential singularities,

99;
of multiform function to obtain branches,

151.
Continuity of a function, region of (see region

of continuity).
Continuous Group, 584.
Contraction of areas in conformal representation,

537.

Convex curve, area of, represented on half-
plane, deduced as the limit of the representa
tion of a rectilinear polygon, 548.
Convex normal polygon for division of plane, in
connection with an infinite group, 595 ;

angles at corners of second category and

of third category, 597 ;
sum of angles at the corners in a cycle
of the first category is a submultiple
of four right angles, 598 ;
when given leads to group, 600 ;
changed into a closed surface, 608.
Corners, of regions, 591 ;

three categories of, for Fuchsian group,

592;

cycles of homologous, 593 ;
how obtained, 596 ;
closed, and open, 596 ;
categories of cycles, 596 ;
of first category are fixed points of

elliptic substitutions, 600 ;
of second and third categories are fixed
points of parabolic substitutions, 600 ;

668

GENERAL INDEX

sub-categories of cycles of, 607 ;

open cycles of, do not occur with Klein-

ian groups, 613.

Crescent changed into another of the same
angle by a linear substitution, 514 ;

represented on a half-plane, 554.
Criterion of character of singularity, 66 ;

class of transcendental integral function,

91.
Critical integer, for expansion of a function in

an infinite series of functions, 124.
Cross-cuts, defined, 314 ;

effect of, on simply connected surface, 316 ;
on any surface, 316 ;
on connectivity of surface, 319 ;
on number of boundaries, 323 ;
and irreducible circuits, 330 ;
on Biemann's surface, 351 ;
chosen for resolution of Eiemann's sur
face, 352 ;
in canonical resolution of Eiemann's

surface, 354 ;
in resolution of Eiemann's surface in its

canonical form, 366 ;
difference of values of integral at opposite

edges of, is constant, 375 ;
moduli of periodicity for, 377 ;

number of independent moduli, 379 ;
introduced in proof of existence-
theorem, 430 et seq.

Curvilinear polygon, bounded by circular arcs,
represented on the half-plane, 549 et seq. ;
function for representation of, 550 ;
equation which secures the representa
tion of, 553 ;

connected with linear differential

equations, 553 ;
bounded by two arcs, 554 ;
bounded by three arcs, 555 (see curvi
linear triangle).

Curvilinear triangles, equation for representa
tion of, on half-plane, 555 ;

connected with solution of differential
equation for the hypergeometric series,
555;

when the orthogonal circle is real, 557 ;
any number of, obtained by inver
sions, lie within the orthogonal
circle, 558 ;

equation is transcendental, 559 ;
discrimination of cases, 559, 560 ;
particular case when the three arcs

touch, 561 ;

when the orthogonal circle is imaginary,
561;

stereographic projection on sphere
so as to give spherical triangle
bounded by great circles, 562 ;
connected with division of spherical
surface by planes of symmetry of
inscribed regular solids, 564 et
seq.;

cases when the relation is algebraical
in both variables and uniform
in one, 564 ;

equations which establish the
representation in these cases,
567 et seq. ;

cases when the relation is algebraical
in both variables but uniform in
neither, 574 et seq.
Cycles of corners, 593 (see corners).
Cyclical interchange of branches of a function
which are affected by a branch-point, 156 ;
when the function is algebraic, 171.

Deficiency of a curve, 356;

is an invariant for rational transforma
tions, 367.

Deformation, of a circuit on a Eiemann's surface
over branch-point impossible, 350 ;

connected surfaces, geometrical and phy
sical, 333 ;

can be effected from one to another
if they be bifacial, be of the same
connectivity, and have the same
number of boundaries, 334 ;
to its canonical form of Eiemann's sur
face with simple winding-points, 365 ;
of loops, 358 et seq. ;
of path of integration, of holomorphic
function does not affect value of the
integral, 26;

over pole of meromorphic function

affects value of the integral, 34 ;
of multiform function (see integral
of multiform function) ;
form of, adopted, 190;
effect of, when there are more

than two periods, 208;
on Eiemann's surface (see path of

integration) ;
of path of variable for multiform

functions, 152;

how far it can take place without
affecting the final branch, 152,
153—155.
Derivative, Schwarzian, 529 (see Schwarzian

derivative).
Derivatives, a holomorphic function possesses

GENERAL INDEX

669

any number of, at points within its region, 32;
superior limit for modulus of, 33 ;
do not necessarily exist along the boun
dary of the region of continuity, 32,
133;
of elliptic functions with regard to the

invariants, 265.

Description of closed curve, positive and nega
tive directions of, 3.

Differential equation of first order, satisfied by
uniform doubly-periodic functions, 237;

in particular, by elliptic functions, 238 ;
satisfied by function which possesses an alge
braic addition-theorem, 309 ;
not containing the independent variable, 470 ;
conditions that integral of, is a uniform
function, 471 et seq. ;

when the integral is uniform, it is
either a rational, a simply-period
ic, or a doubly-periodic, function,
476;

application of general results to bino
mial, 482 et seq.;

discrimination of solutions into the

three classes, 484 et seq. ;
example of integral that is two-valued,

490;

reference to functions which possess ad
dition-theorem, 490.
Dihedral function, automorphic for dihedral

group, 632 (see polyhedral functions).
Dihedral group, of rotations, 623;

of homogeneous substitutions, 624;
of linear substitutions, 625 ;
function automorphic for, 632.
Directions of description of closed curve, 3.
Discontinuous, groups, 584 ;

properly and improperly, 585 ;

all finite groups are, 586 ;

division of plane associated with, 591

(see regions).
Discrete group, 584.

Discriminating circle for uniform function, 111.
Discrimination between accidental and essen
tial singularities, 53, 66.

Discrimination of branches of a function ob
tained by various paths of the variable, 152
—155.
Division of surface into polygons, Lhuilier's

theorem on, 325.
Domain of ordinary point, 52.
Double points of linear substitution, 514.
Double-pyramid, division of surface of circum
scribed sphere by planes of symmetry, 564 ;
equation giving the conformal represen

tation on a half-plane of each triangle
in the stereographic projection of the
divided spherical surface, 567.
Doubly-infinite system of zeros, transcendental

function having, 84.
Doubly-periodic functions, 198;

graphical representation, 199 ;
those considered have only one essential
singularity which is at infinity, 218,
227;
fundamental properties of uniform, 219

et seq. ;
order of, 220 ;
equivalent, 220;
integral of, round parallelogram of

periods, is zero, 221 ;
sum of residues of, for parallelogram, is

zero, 222 ;

of first order do not exist, 223;
of second order consist of two classes,

223;
number of zeros equal to number of

infinities and of level points, 226;
sum of zeros congruent with the sum of
the infinities and with the sum of the
level points, 228 ;

of second order, characteristic equation
of, 231 ;

zeros and infinities of derivative of,

232;

can be expressed in terms of any
assigned homoperiodic function
of the second order with an ap
propriate argument, 223;
of any order with simple infinities can
be expressed in terms of homoperiodic
functions of the second order, 234;
are connected by an algebraical equation

if they have the same periods, 236 ;
differential equation of first order satis
fied by, 237;
in particular, by elliptic functions,

238;

can be expressed rationally in terms of
a homoperiodic function of the second
order and its first derivative, 239;
of second order, properties of (see second

order) ;
expressed in terms of the f-f unction, 256;

and of the <r-function, 260;
possesses algebraical addition-theorem,
299.

Edges of cross-cut, positive and negative, 374,
438.

670

GENERAL INDEX

Edges of regions in division of plane by an
infinite group, 591 ;

two kinds of, for real groups, 592 ;
congruent, are of the same kind, 592 ;
conjugate, 592;
of first kind are even in number and can

be arranged in conjugate pairs, 593;
each pair of conjugate, implies a funda
mental substitution, 593.
Elementary function of second kind, 448 (see

second kind of functions).
Elementary functions of third kind, 449 (see

third kind of functions).
Elementary integrals, of second kind, 396;

determined by an infinity, except as to

additive integral of first kind, 398 ;
number of independent, 399 ;
connected with those of third kind, 403.
Elementary integrals of third kind, 402 ;

connected with integrals of second kind,

403;

number of independent, with same loga
rithmic infinities, 403.
Elements of analytic function, 56;

can be derived from any one when the

function is uniform, 56;
any single one of the, is sufficient for
the construction of the function, 57.
Ellipse, area without, represented on a circle,
501;

area within, represented on a rectangle,
504;

and on a circle, 505.
Ellipsoid conformally represented on plane,

500.

Elliptic functions, obtained by integrating mul
tiform functions, in Jacobian form, 193;

in Weierstrassian form, 196, 249 et seq.;
on a Eiemann's surface, 383 et seq.
Elliptic substitutions, 517, 519 ;

are either periodic or infinitesimal,

521;

occur in connection with cycles of cor
ners, 607, 613.
Equivalent homoperiodic functions, 220;

conditions of equivalence, 225.
Essential singularities, 17, 53 ;

uniform function must assume any value

at, 54, 94;
of transcendental integral function at

infinity, 74 ;

form of function in vicinity of, 96 ;
continuation of function possessing, 99 ;
form of function having finite number
of, as a sum, 100 ;

functions having unlimited number of,

Chap. vn. ;
line of, 140;
lacunary space of, 141 ;
classification of, into classes, 146;
into species, 148;
into wider groups, 148;
of pseudo-automorphic functions, 642;
of automorphic functions, 651.
Essential singularities of groups, 522, 606;

are essential singularities of functions

automorphic for the group, 606;
lie on the fundamental circle, 606 ;
may be the whole of the fundamental

circle, 607.

Existence-theorem for functions on a given
Biemann's surface, Chap. xvn. ;
methods of proof of, 408 ;
abstract of Schwarz's proof of, 409;
results of, relating to classes of functions
proved to exist under conditions, 436.
Expansion of a function in the vicinity of an
ordinary point, by Cauchy's theorem, 43 ;

within a ring, by Laurent's theorem, 47.
Expression of uniform function, in vicinity of
ordinary point, 43;

in vicinity of a zero, 61 ;
in vicinity of accidental singularity, 64 ;
in vicinity of essential singularity, 96 ;
having finite number of essential singu
larities, as a sum, 100;

as a product when without acciden
tal singularities and zeros, 104 ;
as a product, with any number of
zeros and no accidental singu
larities, 108 ;

as a product, with any number of
zeros and of accidental singulari
ties, 110;

in the vicinity of any one of an infinite
number of essential singularities, 113 ;
having an assigned infinite number of
singularities over the plane, 115;

generalised, 116;
having infinity as its single essential

singularity, 117;

having unlimited singularities distrib
uted over a finite circle, 117.
Expression of multiform function in the vicin
ity of branch-point, 158.

Factor, generalising, of transcendental integral
function, 81 ;
primary, 82;
fractional, for potential-function, 422.

GENERAL INDEX

671

Factorial functions, pseudo-periodic on a Eie-
mann's surface, 464;

their argument, 464;
constant factors (or multipliers) for cross
cuts of, 465 ;

forms of, when cross-cuts are canon
ical, 466;

general form of, 466 ;
expression of, in terms of normal ele
mentary functions of the third kind,
466 et seq. ;

zeros and infinities of, 468 ;
cross-cut multipliers and an assigned
number of infinities determine a
limited number of independent, 470.
Factorial periodicity, 586.
Factors (or multipliers) of factorial functions
at cross-cuts, 465 ;

forms of, when cross-cuts are canonical,

466.
Families of groups, seven, 606 ;

for one set, the whole line conserved by
the group is a line of essential singu
larity ; for the other set, only parts of
the conserved line are lines of es
sential singularity, 607.

Finite groups of linear substitutions, 586, 620;
containing a single fundamental substi
tution, 586 ;
anharmonic, containing two elliptic

fundamental substitutions, 587.
Finite number of essential singularities, func
tion having, expressed as a sum, 100.
First kind of pseudo-periodic function, 273.
First kind, of functions on a Eiemann's surface,
436;

moduli of periodicity of functions of,
439 et seq.;

relation between, and those of a func
tion of second kind, 442 ;
when the functions are normal, 447 ;
number of linearly independent functions

of, 443 ;

normal functions of, 446 ;
inversion of, leading to multiply periodic

functions, 453;

derivatives of, as algebraical functions,
461;

infinities and zeros of, 461.
First kind of integrals on Eiemann's surface, 394 ;
number of, linearly independent in

particular case, 395;
are not uniform functions, 395 ;
general vahie of, 396; (see first kind of
functions).

Fixed circle of elliptic Kleinian substitution,

when the equation is generalised, 613.
Fixed points of linear substitution, 514.
Form of argument for given value of function

possessing an addition-theorem, 300 et seq.
Fractional factor for potential function, 422.
Fractional part of doubly-periodic function,

220.
Fuchsian functions, 619 (see automorphie

functions).
Fuchsian group, 591, 606;

if real, conserves axis of real quantities,

591;

when real, it is transformed by one
complex substitution and then con
serves a circle, 603 ;

division of plane into two .portions
within and without the fundamental
circle, 603;
families of, 606;
class of, 608.

Function, Eiemann's general definition of, 8 ;
relations between real and imaginary

parts of, 9 ;
equations satisfied by real and imaginary

parts of, 11 ;
monogenic, defined, 14;
uniform, multiform, defined, 15 ;
branch, and branch-point, of a, defined,

15;

holomorphic, defined, 15;
meromorphic, defined, 16;
continuation of a, 55 ;
region of continuity of, 55 ;
element of, 56 ;

monogenic analytic, definition of, 56 ;
constant along a line or area, if uniform,

is constant everywhere, 59 ;
properties of uniform, without essential

singularities, Chap. iv. ;
integral algebraical, 70 ;
integral transcendental, 70;
having a finite number of branches is a

root of an algebraical equation, 175 ;
potential, 407 (see potential function).
Function possessing an algebraic addition-
theorem, is either algebraic, or algebraic
simply-periodic, or algebraic doubly-periodic,
300;

has only a finite number of values for

one value of the argument, 308 ;
if uniform, then either rational, or
simply-periodic or doubly-periodic, 308 ;
' satisfies a differential equation between
itself and its first derivative, 309.

672

GENERAL INDEX

Functional dependence of complex variables,
form of, adopted, 7 ;

analytical conditions for, 7 ;
establishes conformal representation, 11.
Functionality, monogenic, not coextensive with

arithmetical expression, 139.
Functions, expression in series of (see series of

functions).
Fundamental circle of Fuchsian group, 603 ;

divides plane into two parts which are
inverses of each other with regard to
the circle, 604 ;
essential singularities of the group lie

on, 606.

Fundamental equation for a Riemann's surface
is determined by algebraical functions that
exist on the surface, 462.

Fundamental parallelogram for double period
icity, 200, 206 ;

is not unique, 206.

Fundamental region (or polygon) for division
of plane associated with a discontinuous
group, 591 ;

can be taken so as to have edges of the
first kind cutting the conserved line
orthogonally, 594, 604;

in this case, called a normal polygon,
594;
which can be taken as convex,

595;
angles of, 597 (see convex normal

polygon) ;

characteristics of, 599.
Fundamental set of loops, 360.
Fundamental substitutions of a group, 583 ;
relations between, 584, 593, 599;
one for each pair of conjugate edges of

region, 593.

Fundamental systems of isothermal curves, 579 ;

given by a uniform algebraic function,

or a uniform simply-periodic function,

or a uniform doubly-periodic function,

579;

all families of algebraic isothermal curves
are derived from, by algebraic equa
tions, 580.

General conditions for potential function, 410

(see potential function).

Generalised equations of Kleinian group, 612
(see Kleinian group) ;

polyhedral division of space in connec
tion with, 614 ;

connected with polygonal division
of plane by the group, 615.

Generalising factor of transcendental integral

function, 81.

Graphical determination of, order of infinity of
an algebraic function, 164;

the leading term of a branch in the
vicinity of an ordinary point of the
coefficients of the equation, 167;
the branches of an algebraic function in

the vicinity of a branch-point, 170.
Graphical representation of periodicity of func
tions, 198, 199.

Group of linear substitutions, 582 ;
fundamental substitutions of, 583 ;

relations between, 584 ;
continuous, and discontinuous (or discrete),

584;

properly and improperly discontinuous, 585 ;
finite, 586 (see finite groups) ;
modular, with two fundamental substitu
tions, 587 ;

division of plane into polygons associated
with, 588 et seq. ;

relation between the fundamental

substitutions, 590;

division of plane for any discontinuous group,
591 (see region) ;

fundamental region for, 591 ;
Fuchsian, 591, 606 (see Fuchsian group);
when real, conserves axis of real quantities,

591;

fundamental substitutions of, connected with
the pairs of conjugate edges of a region, 593 ;
seven families of, 606;
conserved line in relation to the essential

singularities, 607;
Kleinian, 610 (see Kleinian group) ;
dihedral, 625 ;
tetrahedral, 627.

Grouping of branches of algebraical function
at a branch-point, 171.

Half-plane represented on a circle, 506 ;
on a semicircle, 506 ;
on a sector, 507 ;
on an infinitely long strip, 508 ;
on a rectilinear polygon, 538 et seq. (see

rectilinear polygon) ;
on a curvilinear polygon, bounded by cir
cular arcs, 549 et seq. (see curvilinear
polygon, curvilinear triangle).
Hermite's sections for integrals of uniform

functions, 185.

Hole in surface, effect of making, on connec
tivity, 320. -
Holomorphic function, defined, 15 ;

GENERAL INDEX

673

integral of, round a simple curve, 23 ;
along a line, 24 ;
when line is deformed, 26 ;
when simple curve is deformed, 27 ;
has a derivative for points within, but
not necessarily on the boundary of,
its region, 32 ;
superior limit for modulus of derivatives

of, 33 ;

expansion of, in the domain of an ordi
nary point, 43, 52 ;
within a ring of convergence by

Laurent's theorem, 47.

Homogeneous form of linear substitutions, 622.
Homogeneous substitutions, 622 ;

two derived from each linear substi
tution, 622 ;
dihedral group of, 624.

Homographic substitution connected with sphe
rical rotation, 620.
Homographic transformation, or substitution,

512 (see linear substitution).
Homologous points, 200, 591.
Homoperiodic functions, 224 ;

when in a constant ratio, 224 ;

when equivalent, 225 ;

are connected by an algebraical equation,

236.
Hyperbolic substitutions, 517, 519 ;

neither periodic nor infinitesimal, 522 ;
do not occur in connection with cycles

of corners, 607, 614.

Hypergeometric series, solution of differential
equation for, connected with couformal repre
sentation of curvilinear triangle, 555 et seq. ;
cases of algebraical solution, 567 et seq.

Icosahedral (and dodecahedral) division of sur
face of circumscribed sphere, 565 ;

equation giving the conformal represent
ation on a half-plane of each triangle
in the stereographic projection of the
divided surface, 573.
Identical substitution, 583.
Imaginary parts of functions, how related to
real parts, 9 ;

equations satisfied by real and, 11.
Improperly discontinuous groups, 585 ;

example of, 615 et seq.
Index of a composite substitution, 583 ;

not entirely determinate, 584.
Infinite circle, integral of any function round,

36.

Infinitesimal curve, integral of any function
round, 35.

F.

Infinitesimal substitution, 584.
Infinities, of a function defined, 16 ;

of algebraic function, 163.

Infinities of doubly-periodic functions, irre
ducible, are in number equal to the irreducible
zeros, 227 ;

and, in sum, are congruent with their

sums, 228 ;
of pseudo-periodic functions (see second

kind, third kind).
Infinities of potential function on a Eiemann's

surface, 435.
Integral function, algebraical, 70 ;

transcendental, 70.

Integral with complex variables, defined, 18 ;
elementary properties of, 19, 20 ;
over area changed into integral round
boundary, by Riemann's fundamental
lemma, 21 ;
of holomorphic function round simple

curve is zero, 23 ;
of holomorphic function along a line is

holomorphic, 24 ;

of meromorphic function round simple
curve containing one simple pole, 27 ;
round simple curve, containing seve
ral simple poles, 28 ;
round curve containing multiple

pole, 32 ;

of any function round infinitesimal circle,
35;

round infinitely great circle, 36 ;
round any curve enclosing all the

branch-points, 37 ;

of uniform function along any line, 184.

Integral of multiform function, between two

points is unaltered for deformation of path

not crossing a branch-point or an infinity, 181 ;

round a curve containing branch-points

and infinities is unaltered when the

curve is deformed to loops, 182 ;

also when the curve is otherwise deformed

under conditions, 183 ;
round a small curve enclosing a branch
point, 183 ;
round a loop, 189 ;
deformed path adopted for, 190 ;
with more periods than two, can be
made to assume any value by modi
fying the path of integration between
the limits, 208.

Integral of uniform function round parallelo
gram of periods, is zero when function is
doubly-periodic, 221 ;

general expression for, 222.

43

674

GENERAL INDEX

Integrals, at opposite edges of cross-cut, values
of, differ by a constant, 375 ;

when cross-cuts are canonical, 377 ;
discontinuities of, excluded on a Eie-

mann's surface, 378 ;
general value of, on a Eiemann's surface,

379;
of algebraic functions, 387 ;

when branch-points are simple, 389 ;
infinities of, of algebraic functions, 390;
first kind of, 394 ;

number of independent, of first kind,

395;
are not uniform functions of position,

395;

general value of, 396 ;
second kind of, 396 (see second kind) ;
elementary, of second kind, 396 (see

elementary integrals) ;
third kind of, 400 (see third kind) ;

elementary, of third kind, 402 (see

elementary integral) ;
connected with integrals of second

kind, 403.
Integration, Eiemann's fundamental lemma in,

21.

Interchange, cyclical, of branches of a function
affected by a branch-point, 156 ;
of algebraical function, 171.
Interchange of argument and parameter in
normal elementary function of the third
kind, 453.
Interchange, sequence of, along branch-lines

determined, 341.

Interchangeable substitutions, 586.
Invariants, derivatives of elliptic functions with

regard to the, 265.
Inversion problem, 455 ;

of functions of the first kind with several
variables leading to multiply periodic
functions, 453 et seq.

Inversions at circles, even number of, lead to
lineo-linear relation between initial and final
points, 523.
Irreducible circuits, 327 ;

complete system contains same number

of, 328 ;
cannot be drawn on a simply connected

surface, 329 ;

round two branch-points, 350.
Irreducible, points, 199, 200, 591, 638 ;

zeros of doubly-periodic function are the
same in number as irreducible infini
ties, 226 ;

likewise the number of level-points, 227 ;
also of automorphic functions, 651 ;
sum of irreducible points is independent

of the value of the doubly-periodic

function, 228.

Isothermal curves, families of plane algebraical,
576;

form of equation that gives such families

as the conformal representation of

parallel straight lines, 579 ;
three fundamental systems of, 579 ;
all, are conformal representations of

fundamental systems by algebraical

equations, 580 ;
isolated may be algebraical by other

relations, 581.

Kinds of edges in region for Fuchsian group,

592 ; (see edges).

Kinds of pseudo-periodic functions, three prin
cipal, 273, 274 ;

examples of other, 295.
Kleiniau functions, 619 ; (see automorphic

functions).
Kleinian group, 610 ;

conserves no fundamental line, 610 ;
generalised equations of, applied to space,
612;

conserve the plane of the complex

variable, 612 ;
double (or fixed) circle of elliptic

substitution of, 613 ;
polygonal division of plane by, 613 ;
polyhedral division of space in connec
tion with generalised equations of, 614 ;
relation between polygonal division of
plane and polyhedral division of space
associated with, 615.

Lacunary functions, 141.

Laguerre's criterion of class of transcendental

integral function, 91.
Lame's differential equation, 281 ;

can be integrated by secondary periodic

functions, 283 ;

general solution for integer value of n,
284;

special cases of ?i=l and n=2, 285.
Laurent's theorem on the expansion of a func
tion which converges within a ring, 47.
Leading term of a branch in vicinity of an
ordinary point of the coefficients of the
equation determined, 167.

Lhuilier's theorem on division of connected
surface into polygons, 325.

GENERAL INDEX

675

Limit, natural, of a power-series, 129.
Linear differential equations of the second
order, connected with automorphic functions,
656.
Linear substitution, 512 ;

equivalent to two translations, a reflexion

and an inversion, 512 ;
changes straight lines and circles into

circles in general, 513 ;
can be chosen so as to transform any

circle into any other circle, 514 ;
changes a plane crescent into another of

the same angle, 514 ;
fixed points of, 514 ;
multiplier of, 515 ;
condition of periodicity, 515 ;
parabolic, 517 ;

and real, 518 ;
elliptic, 517 ;

and real, 519 ;

is either periodic or infinitesimal,

521;
hyperbolic, 517 ;

and real, 519 ;
loxodromic, 517, 521 ;
can be obtained by any number of pairs

of inversions at circles, 523 ;
group of, 582 et seq. (see group) ;
normal form of, 582 ;
identical, 583 ;

algebraical symbols to represent, 583 ;
index of composite, 583 ;
infinitesimal, 584 ;
interchangeable, 586 ;
in homogeneous form, 622.
Logarithmic infinities, integral of third kind
on a Eiemann's surface must possess at
least two, 402.
Loop-cuts, defined, 315;

changed into a cross-cut, 320 ;
effect of, on connectivity, 320 ;

on number of boundaries, 324.
Loops, defined, 153 ;

effect of a loop, is unique, 155 ;
symbol to represent effect of, 357 ;

change of, when loop is deformed,

358;

fundamental set of, 360 ;
simple cycle of, 360 ;
canonical form of complete system of

simple, 361.
Loxodromic substitutions, 517, 521 ;

neither periodic nor infinitesimal, 522 ;
do not occur in connection with cycles
of corners, 613.

Magnification in conformal representation, 11,
492;

in star-maps, 499.
Maps, 499.

Maximum and minimum values of potential
function for a region lie on its boundary, 422.
Mercator's projection of sphere, 498.
Meromorphic function, defined, 16 ;

integral unchanged by deformation of
simple curve in part of plane where
function is uniform, 27 ;
integral round a simple curve, containing

one simple pole, 27 ;
round a curve containing several

simple poles, 28 ;
round a curve containing multiple

pole, 32 ;
cannot, without change, be deformed

across pole, 34 ;

is form of uniform function with a
limited number of accidental singu
larities, 71 ;
all singularities of uniform algebraical,

are accidental, 73.

Mittag-Leffler's theorems on functions having
an unlimited number of singularities, dis
tributed over the whole plane, 112;

distributed over a finite circle, 117.
Modular-function defined, 633 ;

connected with elliptic quarter-periods,

633;

(see modular group) ;
as automorphic functions, 657.
Modular group of substitutions, 587 ;

is improperly discontinuous for real

variables, 585 ;

division of plane into polygons, asso
ciated with, 588 et seq. ;
relation between the fundamental sub
stitutions of, 590 ;

for modulus of elliptic integral, 635 ;
for the absolute invariant of an elliptic

function, 637.
Moduli of periodicity, for cross-cuts, 377 ;

values of, for canonical cross-cuts, 377 ;
number of linearly independent on a

surface, 379 ;
examples of, 379 et seq. ;
introduced in proof of existence-theorem,

430 et seq. ;
of function of first kind on a Biemann's

surface, 439 et seq. ;
relation between, of a function of first
kind and a function of second kind,
442

676

GENERAL INDEX

properties of, for normal function of

first kind, 446 ;

of normal elementary function of second
kind are algebraic functions of its
infinity, 449 ;

of normal elementary function of third
kind are expressed as normal functions
of first kind of its two infinities, 451.
Modulus of variable, 3.
Monogenic, defined, 14 ;

function has any number of derivatives,

14;

analytic function, 56.
Monogenic functionality not coextensive with

arithmetical expression, 139.
Multiform function, defined, 15 ;

elements of, in continuation, 56 ;
expression of, in vicinity of a branch
point, 158 ;
defined by algebraic equation, 161 (see

algebraic function) ;
integral of (see integral of multiform

function) ;

is uniform on Eiemann's surface, 337, 343.
Multiple circuits, 327.
Multiple periodicity, 208 ;

of uniform function of several variables,

209.

Multiplication-theorem, 297.
Multiplicity of zero, 61 ;

of pole, 65.

Multiplier of linear substitution, 515.
Multipliers of factorial functions at cross-cuts,
465;

forms of, when cross-cuts are canonical,

466.

Multiply connected surface, 314 ;
defined, 315 ;

connectivity modified by cross-cuts, 319 ;
by loop-cuts, 320 ;
and by slit, 321 ;

boundaries of, affected by cross-cuts, 323 ;
relation between boundaries of, and con
nectivity, 324 ;
divided into polygons, Lhuilier's theorem

on, 325 ;
number of circuits in complete system

of circuits on, 330.

Multiply-periodic uniform functions of n vari
ables, cannot have more than 2n periods, 209 ;
obtained by inversion of functions of
first kind, 453 et seq.

Natural limit, of a power-series, 129 ;
of part of plane, 558 ;

for pseudo-automorphic function with

certain families of groups, 643.
Negative edge of cross-cut, 374, 438.
Neumann's sphere used to represent the vari
able, 4 ;

used for multiform functions, 153.
Normal elementary function of second kind,

448 (see second kind of functions).
Normal elementary function of third kind, 450

(see third kind of functions).
Normal form of linear substitution, 582.
Normal functions of first kind, 446 (see first

kind of functions).
Normal polygon for division of plane, 594 ;

can be taken convex, 595 (see convex

normal polygon).

Normal surface, Klein's, as a surface of refer
ence of given connectivity and number of
boundaries, 334, 365.

Number of zeros of uniform function in any
area, 61, 63, 68, 72 ;

of periodic functions (see doubly-periodic

functions, second kind, third kind) ;
of pseudo-automorphic functions (see
pseudo-automorphic functions).

Octahedral (and cubic) division of surface of
circumscribed sphere, 565 ;

equation giving the conformal repre
sentation on a half-plane of each
triangle in the stereographic projec
tion of the divided surface, 570.
Open cycles of corners in normal polygon for
division of plane by Fuchsian group, 596
(see corners) ;

do not occur in division of plane by

Kleinian group, 613.
Order of doubly-periodic function, 220.
Order of infinity of a multiform function deter
mined, 164.

Ordinary point of a function, 52 ;
domain of, 52.

Parabola, area without, represented on a circle,
505;

area within, represented on a circle, 506.
Parabolic substitutions, 517, 518;

neither periodic nor infinitesimal, 522 ;
occur in connection with cycles of cor
ners, 607, 613.

Parallelogram for double periodicity, funda
mental, 200, 206;

edges and corners in relation to zeros
and to accidental singularities of func
tions, 218;

GENERAL INDEX

677

is fundamental for a function of the

second order within it, 224.
Parametric integer of thetafuchsian functions,

650.
Path of integration, 18;

can be deformed in region of holomor-
phic function without affecting the
value of the integral, 26 ;
on a Eiemann's surface, can be de
formed except over a discontinuity,
373;

and not over a branch-point, 350.
Periodic linear substitutions, 515 ;

are elliptic, 519.

Periodicity of uniform functions, of one variable,
198 et seq. ;

of several variables, 209.
Periodicity, modulus of, 377 (see moduli of

periodicity).
Periods of a function of one variable, 198 ;

cannot have a real ratio when the func
tion is uniform, 200 ;
cannot exceed two in number indepen
dent of one another if function be
uniform, 205.
Plane used to represent variation of complex

variable, 2.

Poles of a function defined, 16, 53.
Polyhedral division of space in connection with
generalised equations of group of Kleinian
substitutions, 614.

Polyhedral functions, connected with conformal
representation, 566 et seq. ;

for double-pyramid, 567, 632;
for tetrahedron, 568, 630;
for octahedron and cube, 570;
for icosahedron and dodecahedron, 573.
Position on Eiemann's surface, most general
uniform function of, 369 ;

their algebraical expression, 371;
has as many zeros as infinities, 372.
Positive edge of cross-cut, 374, 438.
Potential function, real, defined, 407;

conditions satisfied by, when derived
from a function of position on a Eie
mann's surface, 407 ;
general conditions assigned to, 410;
boundary conditions assigned to, 410 ;
Green's integral-theorems connected

with, 411 et seq. ;

is uniquely determined for a circle by
general conditions and continuous
finite boundary values, 414;

integral expression obtained for,
satisfies the conditions, 417;

the boundary values for circle may
have finite discontinuities at a
limited number of isolated points,
418;

properties of, for a circle, 421 ;
maximum and minimum values of, in a

region, lie on the boundary, 422;
is determined by general conditions and
boundary values, for area conformally
representable on area of a circle, 423;
for combination of areas when it
can be obtained for each sepa
rately, 425 ;
for area containing a winding-point,

428;
for any simply connected surface,

429;

introduction of cross-cut moduli for, on
a doubly connected surface, 430 ;
on a triply connected surface, 433 ;
on any multiply connected surface,

434;

number of linearly independent, every
where finite, 434, 445 ;
introduction of assigned infinities, 435 ;
classes of, determined, 436;
classes of complex functions derived from,

with the respective conditions, 436.
Power- series, as elements of an analytical
function, 56 et seq.; 128 et seq. ;

region of continuity of, consists of one
connected part, 128 ;

may have a natural limit, 129.
Primary factor, 82.

Primitive parallelogram of periods, 206.
Product-form of transcendental integral func
tion with infinite number of zeros over whole
plane, 80.

Pseudo-automorphic functions, 643 (see theta
fuchsian functions).
Pseudo-periodic functions, Chap. xn. ;
of the first kind, 273;
of the second kind, 274;

properties of (see second kind) ;
of the third kind, 274;

properties of (see third kind) ;
on a Eiemann's surface (see factorial

functions).

Pseudo-periodicity of the f- function, 255;
of the (T-function, 260.

Quadrilateral, area of, represented on half-
plane, 546 ;

determination of fourth angular point,
three being arbitrarily assigned, 547.

678

GENERAL INDEX

Ramification of a Eiemann's surface, 349.
Eatio of periods of uniform periodic function

cannot be real, 200.
Rational points in an area, 141.
Real and imaginary parts of functions, how
related, 9 ;

equations satisfied by, 11;
each can be deduced from the other, 12.
Real potential function, 407 (see potential

function).

Real substitutions, 591 (see Fuchsian group).
Reconcileable circuits, 327.
Rectangle, area within, represented on a circle,
502;

and on an ellipse, 504 ;
on a half-plane, 544, 545.

Rectilinear polygon, convex, represented on
half-plane, 538 et seq. ;

function for representation of, 540 ;
equation which secures the representa
tion of, 541 ;

three angular points (but not more) may
be arbitrarily assigned in the repre
sentation, 542 ;

determination of fourth for quadri
lateral, 547 ;

three sides, 543 (see triangle) ;
four sides, 544 (see rectangle, square,

quadrilateral) ;

limit in the form of a convex curve, 548.
Reducible circuits, 327.
Reducible points, 199, 200.
Region of continuity, of a uniform function, .
55, 126 ;

bounded by the singularities, 56;
of a power-series consists of one con
nected part, 128 ;

may have a natural limit, 129 ;
of a series of uniform functions, 132 et

seq. ;

of multiform function, 150.
Regions in division of plane associated with
discontinuous group :
fundamental, 591 ;

uniform correspondence between, 591 ;
contiguous, 591 ;
edges of, 591 (see edges) ;
corners of, 591 (see corners).
Regular in vicinity of ordinary point, function

is, 52.

Regular polygon, area of, conformally repre
sented on a circle, 548 (Ex.).
Regular singularities of algebraical functions,

163.
Regular solids, planes of symmetry of, dividing

the surface of the circumscribed sphere, 564
et seq.
Representation, conformal, 11 (see conformal

representation).

Representation of complex variable on a plane,
2;

and on Neumann's sphere, 4.
Residue of function, defined, 42 ;

when the function is doubly-periodic, the

sum of its residues is zero, 223.
Resolution of Riemann's surface, 351 et seq. ;
how to choose cross-cuts for, 352 ;
canonical, 355;

when in its canonical form, 366.
Revolution, surface of, conformally represented

on a plane, 496.

Riemann's definition of function, 8.
Riemann's fundamental lemma in integration,

21.

Riemann's surface, aggregate of plane sheets,
336;

used to represent algebraic functions, 338;
sheets of, joined along branch-lines, 339;
can be taken in spherical form, 346 ;
connectivity of, with one boundary, 347;

with several boundaries, 350 ;
class of, 34!) ;
ramification of, 349 ;
irreducible circuits on, 350;
resolution of, by cross-cuts into a simply

connected surface, 351 et seq. ;
canonical resolution of, 355 ;
form of, when branch-points are simple,
364;

deformation to canonical form of,

365;

resolution of, in canonical -form, 366;
uniform functions of position on, 369 ;
their expression and the equation

satisfied by them, 371 ;
have as many zeros as infinities,

372;
integrals of algebraic functions on a, 375

et seq.;
existence-theorem for functions on a

given, 405;

functions on (see first kind, second kind,
third kind of functions, algebraic
functions on a).
Riemann-Roch's theorem on algebraic functions

having assigned infinities, 459.
Riemann's theorem on conformal representation
of any plane area, simply connected, on area
of a circle, 526.
Roots of a function, defined, 16,

GENERAL INDEX

C79

Eotations, connected with linear substitutions,
621;

groups of for regular solids, 623 ;
dihedral group of, 623 ;
tetrahedral group of, 625.

Schwarz's symmetric continuation, 57.
Schwarzian derivative, used in conformal re
presentation, 529, 550 et seq.
Second kind of pseudo-periodic function, 274 ;
Hermite's expression for, 277, 279;

limiting form of, when function is
periodic of the first kind, 278, 280;
Mittag-Leffler's expression for, in inter
mediate case, 279, 280;
number of irreducible infinities same as

the number of irreducible zeros, 280 ;
difference between sum of irreducible
infinities and sum of irreducible zeros,
281;
expressed in terms of the cr-function,

281;

used to solve Lamp's differential equa
tion, 281.

Second order of doubly-periodic functions, (see
also doubly-periodic functions), properties of,
Chap. xi. ;

of second class and odd, 243 ;

connected with Jacobian elliptic

functions, 246;
addition-theorem for, 247 ;
of first class and even, illustrated by
Weierstrassian elliptic functions, 249
et seq. ;

of second class and even, 267 et seq.
Second kind, of functions on a Eiemann's sur
face, 436;

relation between moduli of periodicity of
functions of, and those of a function
of first kind, 442 ;
elementary function of, is determined by

its infinity and moduli, 448 ;
normal elementary function of, 448 ;
moduli of periodicity of, 449 ;
used to construct algebraic functions

on a Eiemann's surface, 457.
Second kind, of integrals on a Eiemann's sur
face, 396;

elementary integrals of, 396;
general value of, 398 ;
elementary integrals of, determined by
its infinity except as to integral of
first kind, 398;

number of, 399 ;
(see second kind of functions) ;

two distinct forms of characteristic

equation, 271 ;
compared with elliptic functions,

272.
Secondary periodic functions, 275 (see second

kind).
Sections for integrals of uniform functions,

Hermite's, 185.

Sector on a half- plane, 507 (Ex.).
Semicircle represented on a half-plane, 506 ;

on a circle, 507.
Sequence of interchange along branch-lines

determined, 341.

Series of functions, expansion in, 115 ;
region of continuity of, 132 ;
represents the same function throughout
any connected part of its region of
continuity, 132 ;

may represent different functions in dis
tinct parts of its region of continuity,
137.
Series of powers, expansion in, 43 et seq. ;

function determined by, is the same
throughout its region of continuity,
128;

natural limit of, 129.
Sheets of a Eiemann's surface, 336 ;

relation between variable and, 338 ;
joined along branch-lines, 339.
Simple branch-points for algebraic function,
174;

number of, 175, 356 ;
in connection with loops, 357 ;
canonical arrangement of, 364.
Simple circuit, 327.
Simple curve, defined, 21 ;

used as boundary, 322.
Simple cycles of loops, 360 ;

number of independent, 361.
Simple element for tertiary periodic functions,
of positive class, 291 ;

of negative class, 293.
Simply connected surface, 313 ;
defined, 315 ;
effect of cross-cut on, 316 ;

and of loop-cut on, 320 ;
circuits drawn on, are reducible, 329 ;
winding surface containing oue winding-
point is a, 348.
Simply infinite system of zeros, function having,

83.
Simply periodic functions, 198 ;

graphical representation, 198, 211 ;
properties of, with an essential singularity
at infinity, 212 et seq.;

680

GENERAL INDEX

when uniform, can be expressed as series

of powers of an exponential, 213 ;
of most elementary form, 215 ;
limited class of, considered, 217 ;
possess algebraical addition-theorem,

298.

Singular line, 140.
Singular points, 16.

Singularities, accidental, 16 (see accidental
singularity) ;

essential, 17 (see essential singularity) ;

discrimination between, 53, 66 ;

bound the region of continuity of the

function, 57 ;
must be possessed by uniform functions,

64;

of algebraical functions, regular, 163.
Singularity of a coefficient of an algebraic equa
tion is an infinity of a branch of the function,
164.
Slit, effect of, on connectivity of surface,

321.

Species of essential singularity, 148.
Sphere conformally represented on a plane,
497;

Mercator's projection, 498 ;
stereographic projection, 498.
Spherical form of Eiemann's surface, 346 ;
related to plane form, 347 ;
is bounded, 347.
Spherical surface with holes, connectivity of,

321.
Spheroid, oblate, conformally represented on

plane, 500.

Square, area within, represented on a circle,
502, 545 ;

on a half-plane, 544, 546 ;

area without, represented on a circle,

545.

Stereographic projection of sphere on plane as
a conformal representation, 498 ;

of curvilinear triangle on the surface of

a sphere, 562.
Straight line changed into a circle by a linear

substitution, 513.

Strip of plane, infinitely long, represented on
half-plane, 508 ;

and on a circle, 508 ;
on a cardioid, 536.

Subcategories of cycles of corners, 607.
Substitution, linear or homographic, 512 (see

linear substitution).

Sum of residues of doubly-periodic function,
relative to a fundamental parallelogram, is
zero, 222.

Surface, connected, 312;

has a boundary assigned, 314, 322, 329;
effect of any number of cross-cuts on, 316 ;
connectivity of, 317 ;

affected by cross-cuts, 319 ;
by loop-cuts, 320;
and by slit, 321 ;
class of, 324 ;

supposed bifacial, not unifacial, 325 ;
Lhuilier's theorem on division of, into

polygons, 325 ;

Eiemann's (see Eiemann's surface).
Symbol for loop, 357;

change of, when loop is deformed, 358.
Symmetric continuation, Schwarz's, 57.
System of branch-lines for a Eiemann's surface,

341.

System of zeros for transcendental function,
simply-infinite, 83;

doubly-infinite, 84;

cannot be triply- infinite arithmetical

series, 88 ;
used to define its class, 89.

Tannery's series of functions representing dif
ferent functions in distinct parts of its region
of continuity, 137.
Tertiary periodic functions, 275 (see third

kind).

Test, analytical, of a branch-point, 157.
Tetrahedral division of surface of circumscribed
sphere, 564;

equation giving the conformal represent
ation on a half-plane of each triangle
in the stereographic projection of the
divided surface, 568.

Tetrahedral function, automorphic for tetra-
hedral group, 630 (see polyhedral functions).
Tetrahedral group, of rotations, 625 ;
of substitutions, 627 ;

in another form, 628;
function automorphic for, 632.
Thetafuchsian functions, 642 ;

exists either only within the fundamen
tal circle, or over whole plane, accord
ing to family of group, 643 ;
their essential singularities, 642 ;
pseudo-automorphic for infinite group,

644;

number of irreducible accidental singu
larities of, 644;

number of irreducible zeros of, 648;
parametric integer for. 650 ;
quotient of two with same parametric
integer is automorphic, 651.

GENERAL INDEX

681

Third kind, of functions on a Kiemann's sur
face, 436;

elementary functions of, 449 ;
normal elementary function of, 450 ;
moduli of periodicity of, 451 ;
interchange of argument and para
meter in, 453 ;
used to construct AppelPs factorial

functions, 466 et seq. ;

Third kind, of integrals on a Eiemann's surface,
400;

sum of logarithmic periods of, is zero,

401;
must have two logarithmic infinities at

least, 402;
elementary integrals of, 402 (see third

kind of functions).
Third kind of pseudo-periodic function, 274 ;

canonical form of characteristic equa
tions, 275 ;

relation between number of irreducible
zeros and number of irreducible infini
ties, 286;

relation between sum of irreducible zeros
and sum of irreducible infinities, 287 ;
expression in terms of <7-function, 288;
of positive class, 288 ;

expressed in terms of simple ele
ments, 290 ;
of negative class, 291 ;

expressed in terms of Appell's ele
ment, 293;
expansion in trigonometrical series,

293.

Three principal classes of functions on a
Biemann's surface, 436 (see first kind, second
kind, third kind, of functions).
Transcendental integral function, 70;

it has 2 = 00 for an essential singu
larity, 74;

with unlimited number of zeros over the
whole plane, in form of a product,
76 et seq.;

most general form of, 80 ;
having simply-infinite system of zeros,

83;

having doubly-infinite system of zeros,
84;

Weierstrass's product form of, 87;
cannot have triply-infinite arithmetical

series of zeros, 88 ;
class of, determined by zeros, 89 ;
simple, of given class, 91.

Transformation, homographic, 512 (see linear
substitution).

F.

Triangle, rectilinear, represented on a half-
plane, 543;

separate cases in which representation is
complete and uniform, 543 ;

curvilinear, represented on a half-
plane, 555 (see curvilinear tri
angle).
Trigonometrical series, expansion of tertiary

periodic functions in, 293.

Triply-infinite arithmetical system of zeros can
not be possessed by transcendental integral
function, 91.

Triply-periodic uniform functions of a single
variable do not exist, 205;

example of this proposition, 386.
Two-sheeted surface, special form of branch-
lines for, 344.

Unifacial Surfaces, 325, 333.
Uniform function, defined, 15.
Uniform function, must assume any value at
an essential singularity, 54, 94 ;

has a unique set of elements in continua
tion, 56 ;
is constant everywhere in its region if

constant over a line or area, 59 ;
number of zeros of, in an area, 63 ;
must assume any assigned value, 64 ;
must have at least one singularity, 64 ;
is algebraical polynomial if only singu
larity be accidental and at infinity,
69;

is rational algebraical and meromorphic
if there be no essential singularity and
a finite number of accidental singulari
ties, 71 ;

transcendental (see transcendental func
tion) ;

Hermite's sections for integrals of, 185 ;
of one variable, that are periodic, 200 et

seq. ;
of several variables that are periodic,

208;

simply- periodic (see simply-periodic uni
form functions) ;

doubly-periodic (see doubly-periodic uni
form functions).

Uniform function of position on a Biemann's
surface, multiform function becomes, 337,
343;

most general, 369 ;

algebraic equation determining, 371 ;
has as many zeros as infinities, 372.
Uniform function, conditions that a, be an
integral of a differential equation of first

44

682 GENERAL INDEX

order not containing the independent variable, periodic functions expressed in terms of,

471 et seq. ; 256 ;

when the conditions are satisfied, it is relation between its parameters and

either a rational, a simply-periodic, or periods, 257 ;

a doubly-periodic, function, 476. its quasi-addition-theorem, 261.

Unlimited number of essential singularities, Weierstrass's product- form for transcendental

functions possessing, Chap. vn. ; integral function, with infinite number of

distributed over the plane, 112 ; zeros over the plane, 80 ;

over a finite circle, 117. with doubly-infinite arithmetic series of

zeros, 87.

Weierstrass's ^-function, 251 ; Winding-point, 346.

is doubly-periodic, 252 ; Winding surface, defined, 346 ;

is of the second order and the first class, portion of, that contains one winding-

253 ; point is simply connected, 348.
its differential equation, 254 ;

its addition-theorem, 262 ; Zeros of doubly -periodic function, irreducible,

derivatives with regard to the invariants are in number equal to the irreducible infini-

and the periods, 265. ties and the irreducible level points, 227 ;

Weierstrass's o--function, 249 ; and in sum are congruent with their

its pseudo-periodicity, 259 ; sums, 228.

periodic functions expressed in terms of, Zeros of uniform function are isolated points,

260 ; 60 ;

its quasi-addition-theorem, 261 ; form of function in vicinity of, 61 ;

differential equation satisfied by, 266 ; in an area, number of, 61, 63, 68, 72 ;

used to construct secondary periodic of transcendental function, when simply-

functions, 281 ; infinite, 83 ;

and tertiary periodic functions, 288. when doubly -infinite, 84 ;

Weierstrass's f-function, 250 ; cannot form triply-infinite arith-

its pseudo -periodicity, 255 ; metical series, 88.

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200'

L.

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1/83

/4 33

U.C. BERKELEY LIBRARIES

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